The big picture: distributions as LM outputs

One sentence: we train language models, with reinforcement learning whose reward is a proper scoring rule, to stop emitting a single answer and instead emit a calibrated probability distribution over answers — and we show this works across multiple-choice QA, open-ended QA, medical diagnosis, forecasting, and continuous regression.

Tabs: each project line is self-contained. 1·MCQ replication is the foundation (replicating Bereket et al.); 2·HotpotQA extends it to open-ended answers with an "Other" bucket; 3·DDXPlus trains against full gold distributions (49 diseases); 4·Forecasting moves to continuous targets (Polymarket, Metaculus) with Beta-distribution outputs; 5·Regression-PPL generalizes to mixture-of-Gaussians and probabilistic programs (pluck); Live status shows what is running right now.

The framework (current, after the 2026-05 meetings)

The model reasons freely, then commits to a structured distribution: a short list of named hypotheses with probabilities, plus a residual Other bucket.

<think> ...reasoning... </think>
<hypotheses>
  <h p="0.40">Pneumonia</h>
  <h p="0.30">Bronchitis</h>
  <other p="0.30"/>
</hypotheses>

Listed hypotheses keep their stated weight; the Other bucket spreads its mass over all unlisted answers using the base LM's prior (the LM-prior backoff):

\[ \hat p(a \mid x) \;=\; \begin{cases} w_i & a = h_i \text{ (listed)}\\[2pt] w_{\text{other}}\cdot p_{\text{LM}}(a \mid x) & \text{otherwise} \end{cases} \]

Training reward is the cross-entropy log-score against the gold distribution $p^*$ (a point mass for single-answer tasks, a full differential for DDXPlus):

\[ R \;=\; \sum_a p^*(a)\,\log\bigl(\hat p(a) + \epsilon\bigr) \]

The pipeline. The reward is a strictly proper scoring rule — its expectation is maximized exactly when the model reports its true beliefs.

Why this matters: no behavioral hacks. Earlier iterations needed hand-tuned penalties (hypothesis-count / min-K constraints, soft-diversity bonuses) to stop the model collapsing to one answer or hiding in the Other bucket. The LM-prior backoff gives Other real semantics instead of a black-hole sink, and empirically the cross-entropy objective alone is stable: the model spreads when gold is spread, and commits when gold is concentrated. Formalizing why this objective avoids collapse is the conceptual core of the paper (open theory item).

The cleanest empirical validation: ProtoQA vs TriviaQA

Same objective, opposite learned behaviors — exactly what a proper scoring rule should produce:

Dataset (gold structure)	# hypotheses	w_other	key metric
ProtoQA (multi-cluster gold → should spread)	stays ≈ 3.4	≈ 0.12	coverage 0.07 → 0.20
TriviaQA (single answer → should commit)	3.0 → 1.0	0.02 → 0.48	p_gold 0.21 → 0.55

Project map

Tab	Task / gold structure	Status	Headline
1 · MCQ replication	MedMCQA A–D, single gold	complete	Reproduced Shepard/Bereket: ECE 0.36→0.03 at equal accuracy
2 · HotpotQA	open-ended QA, single gold	Exp 1–10 done	ECE 0.36→0.046; backoff bug found & fixed; principled backoff ≈ min-K hack
3 · DDXPlus	distribution over 49 diseases	Exp 1–5 + TVD audit	Cross-entropy beats every alternative reward; TVD 0.69 is semantic, not formatting
4 · Forecasting	binary + continuous, market-resolved	Tracks 1–2 done	Beta-format 4B ≈ o1 on Brier; mixtures capture 82% of oracle gain on branching
5 · Regression-PPL	continuous targets, censoring, Fermi	Phase 3 training now	MoG-XML wins 3/5 baselines; pluck (exact-inference PPL) beats MoG on survival

Shared findings across projects

Log score > Brier as a reward, consistently (HotpotQA, forecasting): the log score's unbounded penalty disciplines overconfidence harder.
Pure log score is self-calibrating — even with no corrections, ECE drops ~4–8× from base model on every task tried.
Stability depends on scale and structure: Qwen3-4B needs the Other-bucket semantics (or a diversity term) on open-ended tasks; Qwen 34B avoids collapse naturally.
LoRA ≥ full fine-tuning for this objective: LoRA improves monotonically; full FT peaks (~step 100) then overspreads. Hypotheses: weight decay penalizes absolute norm rather than distance-from-base; harder optimization landscape; forgetting. Distance-to-base / KL-to-base regularization flagged as principled fixes.
Always read rollouts, not just metric tables — every major bug (chat-template no-op, off-list diagnoses, prose-instead-of-XML) was found by reading outputs.

Output-representation comparison (DDXPlus, team-level result)

Representation	KL ↓	Mode acc ↑	ECE ↓	Fmt/exec
CoT (verbalized list)	2.17	0.51	0.41	0.97
DSL (class-hierarchy spec)	1.74	0.43	0.16	0.99
Pyro (writes `def model()`)	2.05	—	—	0.98

DSL wins KL and ECE but trades mode accuracy; Pyro writes flat programs — it never reaches for hierarchical structure or noisy-OR (open diagnosis: few-shot demos vs reward design).

Measurement standards

Total variation distance is reported alongside KL on closed-set tasks because it has two readable interpretations:

\[ \mathrm{TV}(P,Q) = \tfrac12 \sum_x |p(x)-q(x)| \;=\; \sup_A |P(A)-Q(A)| \;=\; \inf_{\text{couplings}} \Pr[X \neq Y] \]

Set view (Scheffé): TV = 0.70 means there is a set of answers where gold places 70% more mass than the model; the maximizing set is $\{x : p(x) > q(x)\}$.
Coupling view: $1-\mathrm{TV} = \sum_x \min(p(x), q(x))$ is the "paint overlap" between the two distributions.

Reporting rules: always give validation reward next to accuracy and ECE (the reward jointly captures calibration + accuracy); report TV alongside KL on closed sets. Caveat: Tim Vieira flagged literature questioning ECE as the right calibration metric — under review.

Matched-subset reporting (project-wide standard, adopted 2026-06-18). When a model can abstain by failing to produce a parseable answer, parse failure is informative: models decline the examples they find hardest. A metric averaged only over parsed outputs is therefore selection-biased, and is not comparable across models or methods with different parse rates (a model that answers only the easy 60% looks better than one that answers all 100%). The standard is to compare on the matched subset — the examples every series parses — and to quote each series' parse rate beside the metric. This first bit us on Regression-PPL (a trained model looked worse than base on fermieval purely because base abstained on the hard questions); the same commit-vs-abstain effect underlies the forecasting Beta-format result. Tooling: matched_report.py.

The broader vision (Alex)

LM outputs as a spectrum of increasingly expressive uncertainty representations:

Aspirational endpoint: the model emits a latent-variable diagnosis simulator; conditioning it on symptoms yields the posterior. Pluck (Alex's exact-marginalization PPL, unlike approximate Pyro/NumPyro/Gen) is the candidate engine. The cross-entropy + Other framework is the ICLR-scoped contribution; this is the long arc.

Glossary

Other bucket	residual-mass token whose probability is distributed over unlisted answers via $p_{\text{LM}}$
LM-prior backoff	using the base model's logit probabilities to spread Other-bucket mass
Coverage	fraction of gold mass captured by listed hypotheses
Mode accuracy	argmax of predicted distribution matches argmax of gold
Proper scoring rule	loss minimized in expectation at the true distribution (here: log-score / cross-entropy)
RLVF	RL with verifiable feedback — the single-correct-answer baseline
GRPO / Dr. GRPO	group-relative policy optimization: sample G completions per prompt, advantage = reward − group mean (Dr. variant: no std normalization)
ECE	expected calibration error — gap between stated confidence and empirical accuracy
Pluck	Alex's probabilistic programming language doing exact marginalization (Rust, Python bindings)

Project 1 — Bereket replication: calibrated MCQ distributions

What it is. The foundation everything else builds on: replicate Bereket et al.'s result (via Shepard's LoRA branch) that GRPO with a log-score reward teaches Qwen3-4B to output a calibrated distribution over the four MedMCQA options instead of a single letter.

<hypotheses>{A: 0.40; B: 0.30; C: 0.20; D: 0.10}</hypotheses>

Reward is $ R = \log p(\text{gold}) $ — the categorical log-score over the four options. Training: GRPO, group size 16, effective batch 512, LoRA r=64 on Qwen3-4B with vLLM colocated for generation.

Results: replication achieved

Metric (4,183 val samples)	Base Qwen3-4B	Shepard LoRA 240s (reference)	Ours: LoRA 5e-6, 450s
Format rate	0.865	0.998	0.994
Accuracy	0.556	0.585	0.580
ECE	0.104	0.019	0.034
Mean log-score	−1.40	−1.04	−1.10

Our configs	Steps	ECE	Acc	Fmt	Log score
LoRA lr=5e-6 (reproduction)	450	0.034	0.580	0.994	−1.100
LoRA lr=1e-5 cosine (3.75× faster)	120	0.032	0.550	0.966	−1.272
Full FT lr=5e-6 (peaks then regresses)	100	0.054	0.553	0.977	−1.233

LoRA vs full fine-tuning (also verified on GSM8K) LoRA improves monotonically with no regression — its implicit regularization prevents overspreading. Full FT peaks around step 100, then accuracy drops and ECE regresses, even with KL regularization. Working hypotheses: weight decay penalizes absolute weight norm rather than distance-from-base; harder optimization landscape; forgetting. Principled alternatives flagged: distance-to-base and KL-to-base regularization.

Hard-won technical lessons (inherited by every later project)

vLLM sleep mode is broken in TRL 1.1.0.dev0 — corrupts weights on wake, producing garbage generations. Always vllm_enable_sleep_mode=False; weights sync in-place instead.
loss = 0 is normal for Dr. GRPO — with advantages centered per group and importance ratios ≈ 1, the loss value collapses to 0; learning happens through the (non-zero) gradients.
Qwen3 thinking mode must be disabled via chat-template patching, or long <think> chains eat the token budget before the distribution is emitted.
Learning-rate window is narrow: LoRA 1e-6 doesn't converge on this TRL version, 5e-6 converges in ~450 steps, 1e-5 cosine in ~90–120, 1.5e-5 goes overconfident.
bf16 off for LoRA (fp32 needed for small-lr adapter updates); bf16 on for full FT (memory).
Zero-advantage filtering: completions whose reward equals the group mean (e.g., when the whole group fails to parse) are excluded from the loss to remove gradient noise.

Code lives at the repo root (train.py, reward.py, patches.py, research_log.md). Status: complete; serves as the baseline-replication anchor for the paper.

Project 2 — Multi-answer RL on HotpotQA: the Other bucket

What it is. The jump from multiple choice to open-ended QA: the answer space is unbounded, so the model can no longer enumerate it. It generates K candidate answers with weights plus an Other bucket for everything unlisted, summing to 1. Dataset: HotpotQA-Modified — 10 context paragraphs (2 gold + 8 distractors), with 0/1/2 gold paragraphs randomly dropped to induce genuine ambiguity. Model: Qwen3-4B, GRPO, 25k train / 500-sample calibration eval.

<hypotheses>
<h1>First candidate</h1><w1>0.40</w1>
<h2>Second candidate</h2><w2>0.30</w2>
<h3>Third candidate</h3><w3>0.20</w3>
<other>0.10</other>
</hypotheses>

Reward semantics. The backoff branch implements the induced distribution $\hat p(\text{gold}) = w_0 \, p_{\text{LM}}(\text{gold}\mid x)$; the intractable denominator $p_{\text{LM}}(\mathcal{Y}\setminus H \mid x)$ is dropped (weak function of H).

Design decisions

Feared failure mode — "w_other=1 as a safe hedge" — never happens. With one listed hypothesis at weight $w$, expected reward is the binary log-score $q\log w + (1-q)\log(1-w)$, maximized at $w=q$: listing your best guess always beats pure Other. The pure-log-score local optimum is therefore "1 hypothesis + Other", which is exactly what collapse experiments show.
Diversity terms (the pre-framework hacks): hard min-K (penalty if fewer than 3 hypotheses) and a smooth soft-diversity variant $\min(0, \log(n/K))$. Superseded in principle by the backoff, but still pragmatically useful (see Exp 9/10).
Equivalence-class matching: token-F1 ≥ 0.75 plus token-contiguous substring rule; all matching hypotheses' weights are summed. The earlier F1 ≥ 0.5 first-match rule produced ~9% spurious matches ("David Diamond" ↔ "David Weissman" via shared "David"); the fix brings this to ~3%, all legitimate variants.
Fail reward ordering is load-bearing: a valid-but-wrong output must never score below a parse failure, or the model learns to abandon the format (observed: format rate 94% → 47% in 21 steps when this was violated).

The backoff bug — and why "read the rollouts" is a project rule

Broken until 2026-04-25: a silent no-op chat-template patch. The precompute script "disabled thinking" via chat_template.replace("enable_thinking=True", "enable_thinking=False") — but the Qwen3 template never contains that literal string, so nothing changed. The base LM was then scored for $p(\text{gold})$ at a position where it expects to open a <think> block: single-token answers like "no" or "Scotland" got log-probs of −30 to −42. Every backoff experiment (Exp 7/8) trained against this pathological prior.

The fix: append a literal "Final answer: " prefix after the chat template before scoring gold. Diagnostic over prompt variants (200 examples, per-token mean $\log p(\text{gold})$):

Prompt variant	per-token mean	median
V0 — broken baseline (assistant + think position)	−15.24	−13.46
V5 — thinking disabled only	−15.80	−12.55
V1 — "Final answer: " prefix (chosen)	−1.53	0.00
V2 — raw QA format	−1.18	0.00

On the full 5k-example precompute: total $\log p$ mean −43.92 → −4.64; per-token −14.66 → −1.33. The "gold missed" reward is now comparable in magnitude to the "gold listed" reward, as the theory wants.

Experiment grid

Exp	Config	Acc	top1 ECE	Brier	# hyps	w_other	Takeaway
Base	Qwen3-4B untrained	0.499	0.361	0.347	1.9	—	overconfident
1	log + CW	—	—	—	1.0	0.43	collapses without diversity
2	log + CW + min-K (best ckpt s600)	0.547	0.046	0.169	3.0	0.30	best overall
3	pure log	—	0.099	—	1.1	0.39	self-calibrating but collapses
4	log + min-K	0.505	0.043	0.181	3.0	0.28	CW adds ~2% acc
5	Brier + CW + min-K	0.515	0.060	0.176	3.0	0.29	log > Brier
6	log + CW + soft-diversity	0.516	0.069	0.181	3.0	0.25	smoothest w_other plateau
7v2	backoff, no min-K (broken precompute)	0.506	0.101	0.189	2.0	0.30	partial collapse
8	backoff + min-K (broken precompute)	0.499	0.083	0.179	2.9	0.26	no gain over min-K
9	fixed backoff, no min-K (5k train)	0.476	0.063	0.175	2.9	0.25	diversity without min-K
10	fixed backoff + min-K (5k train)	0.520*	—	—	2.7	0.22	*training-eval; calib eval pending

Evals at step 300 (Exp 2: step 600) on 500 validation samples, temp 0.6. Exp 9/10 used 5k train samples vs 25k for Exp 2 — scale confound when comparing accuracy.

The headline of Exp 9 (2026-04-25). With the corrected backoff, the model maintains ≈2.9 hypotheses with no min-K hack at all (vs 2.0 under the broken precompute, and 1.0–1.1 with nothing). The principled formulation does the job the hack was doing. Caveat: Exp 10 suggests min-K may still buy accuracy (0.520 vs 0.470 training-eval), and the broken-era conclusion "backoff adds nothing over min-K" needs the scale-matched rerun before it's final.

Calibration must be viewed three ways

The historical metric, top1_ece, looks only at the heaviest hypothesis — it never touches w_other and hid systematic under-confidence. The eval now reports:

top1_ece — heaviest-hypothesis calibration (Exp 2 s600: 0.076)
p_gold_ece — uses w_other when gold is unlisted (same model: 0.424 — large hidden underconfidence)
p_gold_backoff_ece — full induced distribution $w_0 \cdot e^{\text{base-logprob}}$

Open items

Scale-matched Exp 2 vs Exp 9 comparison (5k/300 vs 25k/600 confound).
Exp 10 calibration eval (training done; eval crashed at GPU init).
BPE-boundary outliers: ~2% of answers tokenize as a single merged token at the prompt boundary, distorting the auto-computed fail reward (−42 instead of ≈−10); switch to a percentile.
Jacob's residual (include $w_0\, p_{\text{LM}}$ even when gold is listed) implemented but untrained; semantically cleaner, numerically tiny (+0.036 at $w_0$=0.3).
Other-bucket mass currently double-counts listed answers in $p_{\text{LM}}$ — documented approximation (clean fix needs per-rollout precompute).
Full 7,405-sample validation eval; LoRA run of the best config.

Status: analysis phase — no runs active. Last activity 2026-04-25 (audit + Exp 9/10). Per the 2026-05 meetings, exact-match was reinstated for HotpotQA (fuzzy matching removed → more stable), and the LM-prior backoff is the framework-level formulation going forward.

Project 3 — DDXPlus: training against full gold distributions

What it is. The only task where gold is itself a distribution: DDXPlus (Fansi Tchango et al., NeurIPS 2022) is a synthetic medical-diagnosis benchmark with a closed set of 49 pathologies and a weighted differential diagnosis per patient (median ~8 plausible conditions). Prior work (Puri et al. 2025) collapses the differential to binary set membership; we train Qwen3-4B (GRPO, LoRA) against the actual probability weights — distribution-to-distribution. Patient evidence arrives as codes (E_135_@_V_3) and is mapped to readable symptom descriptions; the system prompt enumerates all 49 condition names (without this the model invents non-matching terms and the reward signal is too weak to learn from).

Design

Reward — cross-entropy against the gold differential (strictly proper; reduces to $\log p(\text{correct})$ when gold is a point mass): \[ R = \sum_{j} p^*_j \,\log\bigl(\max(\hat p_j,\ \epsilon)\bigr) \]
$\epsilon$ must be exactly 1/49 — i.e., a missing condition is scored as if under a uniform prediction ($\log 1/49 = -3.89$ = fail reward). With smaller $\epsilon$, wrong-but-formatted outputs score below a parse failure and the model learns to break the format.
No Other bucket — the set is closed; unlisted conditions get $\epsilon$.
Variable-length output, same <hypotheses> XML as HotpotQA; the model typically lists ~4 conditions (gold has ~9).

Reward ablation: cross-entropy dominates everything

Metric (2,000-sample eval, best ckpt each)	Base	Exp 1 xent	Exp 2 xent+CW	Exp 3 Brier+CW	Exp 4 mode-only	Exp 5 set-coverage
Format rate	0.17	0.987	0.992	0.991	0.995	0.996
Mode accuracy	0.040	0.459	0.508	0.496	0.496	0.473
Top-3 coverage	0.030	0.439	0.456	0.414	0.436	0.298
KL ↓	2.436	1.042	1.012	1.081	1.056	1.217
TVD ↓	0.978	0.690	0.687	0.696	0.703	0.775
ECE ↓	0.190	—	0.114	—	0.133	0.434
Brier ↓	0.203	—	0.043	—	0.052	0.491
# conditions listed	—	4.0	4.0	4.0	4.1	1.3

Paper-ready claims. (1) Cross-entropy + correctness-weight wins on every metric. (2) Puri-style set-coverage reward causes mode collapse: 1.3 conditions at 0.89 max confidence, ECE 0.434 — set membership is fundamentally incompatible with calibration. (3) Mode-only RLVF underperforms distributional training even on mode accuracy itself (0.496 vs 0.508) — the secondary conditions act as a regularizer. This is the baseline that supports the headline claim that distributional training adds value. (4) A 4B model with distributional training matches Puri's 8B set-coverage results (50.8% mode accuracy).

TVD decomposition audit (2026-05-28): is 0.69 real disagreement or formatting noise?

KL ≈ 1.0 with TVD ≈ 0.69 sits essentially on the Pinsker bound $ \mathrm{TV} \le \sqrt{\mathrm{KL}/2} $ — which signals mass-in-wrong-places rather than mild miscalibration. The audit decomposed the per-sample mean TVD by Scheffé sets:

Scheffé / signed-gap aggregation over 2,000 validation patients, best model (Exp 2 step 250).

Reading the 30 highest-TVD patients (all TVD = 1.0) classifies the disagreement into three semantic modes:

1 — Base-rate / familiarity bias (most common)

The model defaults to prototypical internet-medicine respiratory diagnoses regardless of evidence. Over-predicted: Influenza (+4.4pp), Acute rhinosinusitis (+3.3), Viral pharyngitis (+3.2), URTI (+2.5), Pneumonia (+2.0). Under-predicted: Bronchitis (−3.5), Chagas (−3.1), Anemia (−3.1), Acute dystonic reactions (−2.3), Sarcoidosis (−2.0). Example: 6-year-old with sharp chest pain → gold: pulmonary embolism + pneumothorax + myocarditis; predicted: pneumonia + bronchitis + pharyngitis.

2 — Sibling-disease confusion (right organ system, wrong specific)

NSTEMI/STEMI ↔ unstable angina; laryngospasm ↔ croup/epiglottitis; acute otitis media ↔ rhinosinusitis variants. These look like token-overlap canonicalization problems but are clinically distinct diagnoses. This mode makes PE and NSTEMI appear on both sides of the macro ledger — correct rate, wrong patients.

3 — Off-list invention (closed-set violation)

2.7% of predicted mass per sample lands on names outside the 49 (12% of samples have any). Most common: "Heart failure" (2.4%), "Acute pharyngitis" (1.4%), "Psoriatic arthritis" (1.2%). Genuine canonicalization gaps ("Acute pharyngitis" → "Viral pharyngitis", "Myocardial infarction" → "Possible NSTEMI/STEMI") total under 1% of mass — fixing string matching would barely move TVD. Parse failures: 17/2000 (0.85%), all from <think> overrunning the 1,024-token budget.

Verdict for the paper. TVD ≈ 0.69 is not a measurement artifact: it reflects genuine diagnostic disagreement — prior over-shift toward common infections, under-discrimination within disease families, and a 2–3% closed-set violation rate. The levers are better intra-family discrimination and breaking the base-rate prior, not harder string canonicalization.

Open items

Test-split evaluation (134k samples) for final paper numbers; current finals are on 2,000 validation samples.
Explain the DSL representation's mode-accuracy drop (calibration-for-sharpness tradeoff or genuinely worse argmax?) and diagnose why Pyro writes flat programs — see the representation-comparison table in the Overview tab.
Data/model-scale levers against the base-rate prior (25k of 1M+ train examples used; all runs LoRA).
Optional: alias-canonicalization postprocessor (recovers <1pp TVD — probably not worth it).

Status: Exp 1–5 and the TVD audit complete. Artifacts: tvd_decomposition_exp2_s250.json, tvd_per_sample_inspection.md, analyze_tvd.py, inspect_per_sample_tvd.py.

Project 4 — Forecasting: from binary markets to mixture-of-Betas

What it is. The framework's first step beyond discrete answer sets, in two tracks. Track 1 replicates the "turtle paper" (outcome-based RL to predict the future) on Polymarket binary questions: the model reads a question written before resolution and outputs a probability, rewarded by a proper scoring rule once the outcome is known. Track 2 moves to continuous targets, where the model outputs a full density — first a single Beta distribution, then a mixture of Betas. Everything is GRPO + LoRA on Qwen3-4B (scale probes at 14B/32B), trained on 4.7k synthetic "future-as-label" examples, tested on 1,265 real Polymarket questions (zero train/test contamination verified by exact-match and Jaccard checks).

Track 1 — Polymarket binary forecasting

Headline table below is the historical parse-conditioned view — each model scored only on the questions it parsed. The parse % column is the new addition that exposes the catch: the Beta-format methods abstain on ~20% of questions, so their Brier is not comparable to the ~99%-parse point methods. The set-matched comparison directly below corrects this.

Model / config	parse %	Brier ↓	ECE ↓	Acc ↑	extreme (>0.95) %
Base Qwen3-4B, point probability	~0.99	0.486	0.487	0.420	~20%
Base Qwen3-14B, point probability	0.990	0.347	0.262	0.576	34%
Base Qwen3-14B, Beta format, zero-shot	0.809	0.211	0.091	0.679	2%
Base Qwen3-32B, Beta format, zero-shot	0.912	0.204	0.113	0.688	2.4%
4B + Brier-reward LoRA (s300)	0.993	0.256	0.143	0.638	8.2%
4B + log-reward LoRA (s300)	0.997	0.241	0.141	0.648	5.8%
4B + Beta-format RL (s200) — best 4B	0.802	0.213	0.093	0.692	0.2%
o1 (reference, no RL)	1.000	0.202	0.096	0.700	—
ReMax 14B, 110k train (turtle paper SOTA)	1.000	0.191	0.053	0.717	—
Polymarket aggregate odds (oracle)	1.000	0.151	0.045	0.781	—

Brier/ECE/Acc here are each model's own parsed subset (1,265-question turtle test set). Parse % from forecasting_matched.py; Beta-format rows abstain heavily (PP-Beta 4B parses 1014/1265). o1 / ReMax / Polymarket predictions are full-coverage references shipped with the test set.

Findings.

Log > Brier as RL reward (0.241 vs 0.256): the quadratic Brier penalty is too gentle on confident-and-wrong predictions.
The Beta output format is a structural regularizer — but part of the apparent edge is abstention. A 14B base model asked to emit $\mathrm{Beta}(\alpha,\beta)$ with zero training posts 0.211 Brier with 2% extremes. For a binary outcome the Beta marginal likelihood reduces exactly to $\log \mathbb{E}[\theta]$ — identical to a point probability — so any gain is a change in the reasoning pathway, not the scoring. But it parses only 81% of questions: on the set-matched subset (below) the 14B point-vs-Beta gap shrinks from 0.347→0.211 (=0.136) to 0.282→0.200 (=0.082), so ~40% of the "massive" advantage was the Beta format abstaining on the hard questions the point format had to answer.
Bottleneck is training data, not algorithm: the gap to ReMax (0.022 Brier) tracks their 110k vs our 4.7k training examples.
Entropy bonuses and probability-clamping hacks both failed to generalize.

Prompt ablation (2026-05-28) reframes the Beta story. Nine prompt variants on 250 test questions, Qwen3-14B zero-shot, scored with an honest Brier (failed parses charged at the base rate): the raw "Beta wins by −0.150" shrinks to −0.079 once parse failures are charged; simple discrete quantization of output probabilities to {0.05,…,0.95} reaches 0.262 with perfect parse rates, rivaling Beta; and just reordering the prompt to "output FIRST, then reason" captures about half the Beta benefit (0.276) with no format change at all. Interpretation: much of the Beta advantage is commit-before-rationalize behavior plus selective abstention on hard questions — the format matters less than when the number is emitted.

Example: one question, three prompt formats (click to expand)

The same question ("Will Apple launch an iPhone SE on February 19?", gold = No) answered by the 14B base model under three prompt formats — full reasoning and completion each:

One question, three prompt formats — "Will Apple launch an iPhone SE on February 19?" (gold: No (0) (reference probs — Polymarket 0.92, o1 0.30))

log baseline — P(Yes) = 0.600, Brier 0.360

② Model reasoning (<think>) under this prompt format:

Okay, let's tackle this step by step. The question is whether Apple will launch an iPhone SE on February 19, 2025. The resolution criteria say that if they announce it during the event on that date, it's a "Yes," otherwise "No." The event is scheduled by Tim Cook, so that's a given.

Looking at the news articles, there's a mix of information. Some articles suggest the iPhone SE 4 is coming soon. For example, article [1] from MacRumors says it's expected in early 2025, with a possible price starting at $429. Article [2] from AppleInsider mentions the iPhone SE 4 is expected by February 21, with product briefings before the announcement. Article [3] from Tom's Guide says it might be launched as early as this week or later this spring. Article [4] from ZDNet says Apple is expected to launch it this week. Article [5] from MacRumors forums has users speculating it could be announced this week, but some think March. Article [6] from AppleInsider mentions a possible announcement in the week of February 14, with units reaching customers later. Article [7] from 9meters.com says it's expected to be announced on February 11, with pre-sales starting February 14 and full release in late February. However, article [8] from MacWorld says February is slow for Apple, with no mention of an iPhone SE launch on Feb 19, and Mark Gurman suggests holding it for a March event. Articles [9] and [10] also point towards March or April as the likely launch dates, based on past patterns and supply chain reports.

Now, the key date here is February 19. The event is scheduled for that day. But the articles are conflicting. Some say it's coming soon (this week, Feb 11, 14, 21), others suggest March. The resolution criteria state that if the event on Feb 19 doesn't have an announcement, it's a "No." However, the user's current date is Feb 14, so there's only 5 days left until the event.

Looking at the latest articles, [7] says the announcement is on Feb 11, but that's already passed. If that's the case, maybe the event on Feb 19 is for another product. But article [2] says the iPhone SE 4 is expected by Feb 21, which is after the event. However, the event is on Feb 19, so if the announcement is on Feb 21, then the Feb 19 event wouldn't include it. But article [7] also mentions that the announcement is through a press release, not a big event. However, the resolution criteria say that the event on Feb 19 is the one to check.

But there's also article [10] which says March is the likeliest date based on past launches. Previous SE models were launched in March 2022, April 2020, March 2017. So maybe Apple follows a pattern of launching SE in March. However, some articles suggest earlier dates.

But the user's current date is Feb 14, and the event is on Feb 19. If the latest info is that the SE 4 is expected to be announced on Feb 11 (which is past) or in the week of Feb 14, but the event is on Feb 19. If the announcement is scheduled for the Feb 19 event, then it's a "Yes." But if the announcement is scheduled for later (like Feb 21 or March), then it's a "No."

However, article [2] says that Apple is preparing announcements, with the iPhone SE 4 expected by Feb 21. The event is on Feb 19, so if the announcement is on Feb 21, then the Feb 19 event wouldn't include it. But the user's question is about the Feb 19 event.

Also, article [7] says the SE 4 will be announced through a press release, not a big event. But the resolution criteria say that the event on Feb 19 is the one to check. If the announcement is via press release, maybe not during the event. However, the event is scheduled by Tim Cook, so maybe the event is the main announcement.

But there's conflicting info. Some articles suggest the SE 4 is coming soon (this week or early Feb), others suggest March. Given that the event is on Feb 19, and the latest articles (as of Feb 14) have some saying it's expected by Feb 21, but others suggesting March.

Considering the base rates, Apple's past SE launches were in March, April, etc. But the current speculation is that they might be moving it earlier. However, the event on Feb 19 is a specific date. If the latest info is that the SE 4 is expected by Feb 21, then the Feb 19 event might not include it. But if the event is the main one, maybe they'll announce it then.

But the user's question is whether the event on Feb 19 will include the announcement. Given that some articles say the SE 4 is expected by Feb 21, and others say March, but the event is on Feb 19. If the announcement is scheduled for the event, then it's a "Yes," else "No."

However, the resolution criteria state that if the event doesn't occur by Feb 19 11:59 PM ET, it's a "No." But the event is scheduled, so that's not the issue. The key is whether the SE is announced during that event.

Given the conflicting reports, but the fact that some articles (like [7]) suggest the announcement is on Feb 11 (already passed) or pre-sales starting Feb 14, but the event is on Feb 19. If the SE is announced via press release on Feb 11, then the Feb 19 event might not be the one. However, if the event on Feb 19 is the main one, and the SE is announced then, it's a "Yes."

But the latest article [10] from Jan 29 says March is the likeliest date. However, other articles are more recent (Feb 13, 12, etc.) suggesting earlier dates.

Considering the time proximity, the articles from Feb 13 and 12 are more recent. Article [1] (Feb 13) says early 2025, which could be Feb. Article [2] (Feb 12) says expected by Feb 21. Article [7] (Feb 7) says announced on Feb 11, but that's past.

If the event is on Feb 19, and the latest info is that the SE is expected by Feb 21, then it's possible that the Feb 19 event is the one where they announce it. However, if the SE is expected by Feb 21, then the Feb 19 event might not be the one.

But the resolution criteria say that the event on Feb 19 is the one to check. If the SE is announced during that event, it's a "Yes." Otherwise, "No."

Given the conflicting info, but the fact that some articles suggest the SE is coming soon (this week or early Feb), and the event is on Feb 19, which is within the timeframe of some predictions, I think there's a moderate chance. However, other articles suggest March.

Considering the base rate of Apple's past SE launches (March), but the current speculation is that they might be moving it earlier. However, the event on Feb 19 is a specific date. If the latest info is that the SE is expected by Feb 21, then the Feb 19 event might not include it. But if the event is the main one, maybe they'll announce it then.

I think the probability is around 0.5, but considering that some articles are more recent and suggest the SE is coming soon, maybe a bit higher. However, the conflicting info makes it uncertain. So maybe 0.6? But I need to be careful.

Alternatively, since the event is scheduled, and the SE is expected by Feb 21, which is after the event, but the event is on Feb 19. If the SE is announced during the event, then it's a "Yes." But if the announcement is after the event, then "No."

Given that the latest articles (Feb 13) say early 2025, which could include Feb 19. But others say March.

I think the best estimate is around 0.6, but I'm not sure. Maybe 0.55? Or 0.6. Given the conflicting info, but the fact that some articles are pointing towards the event date, I'll go with 0.6.

④ Model completion:

<probability>0.60</probability>

Beta(α,β) format — P(Yes) = 0.600, Brier 0.360

② Model reasoning (<think>) under this prompt format:

Okay, let's tackle this. The question is whether Apple will launch an iPhone SE on February 19. The resolution criteria say that if they announce it during the event on that date, it's a "Yes." The event is scheduled for February 19, and the market will resolve to "No" if it doesn't happen by that date.

Looking at the news articles, there's a mix of information. Some articles suggest the iPhone SE 4 is expected to be announced soon. For example, article [1] from February 13 says the launch is expected in early 2025, possibly around the time of the event. Article [2] from February 12 mentions the iPhone SE 4 is expected by February 21, which is after the event date. Article [3] from February 11 says it might be launched as early as this week or later in spring. Article [4] from February 11 also mentions a possible launch this week. Article [5] from February 10 has users speculating it might be announced this week, but previous SE models were in March. Article [6] from February 7 says it might be announced the week of February 14, which is close to the event date. Article [7] from February 7 mentions an expected announcement on February 11, but that's before the event. Article [8] from February 3 says February is slow for Apple, and the SE might be held for a March event. Article [9] from January 31 suggests a March or April launch. Article [10] from January 29 points to March as the likely date.

So, the conflicting info is that some sources say early February (like the week of 14th or 11th), others say March. The event is on February 19. The key is whether the announcement is scheduled for that exact date. The resolution criteria require an announcement during the event on February 19. However, some articles suggest the SE might be announced before that (like February 11 or 14), while others say March. Also, article [7] says it's announced via press release, not a big event, which might mean it's not during the scheduled event on 19th. But the event is scheduled by Tim Cook, so maybe the announcement is part of that event. However, other sources like [8] and [10] suggest March is more likely.

Considering the dates: the event is on February 19. The latest articles (from Feb 13 and 12) mention the SE 4 expected by Feb 21, which is after the event. But the event is on the 19th. If the announcement is on the 19th, then it would count. However, other articles suggest earlier dates or March. The uncertainty here is that some sources are conflicting. The base rate for Apple launching a new SE model in February is low, as past launches were in March (2022, 2020, 2017). However, the latest rumors are pushing it to early February.

Given that the event is scheduled, and some sources say the SE is expected around that time, but others say March, the probability is moderate. The confidence isn't high because of conflicting reports. So, a Beta distribution with moderate confidence, maybe around 0.6 to 0.7, but with some uncertainty. Let's say alpha=7, beta=3 (mean 0.7, moderate confidence). But considering the conflicting info, maybe lower. Alternatively, if the event is on 19th and some sources say it's expected by 21st, maybe the chance is higher. However, the event is on 19th, and if the announcement is during that event, it's a yes. But if the SE is announced before the event (like on 11th or 14th), then the event might not include it. But the resolution criteria say the event is scheduled, and the market resolves based on the event. So if the SE is announced during the event, it's yes. If not, no. But the articles are conflicting on the exact date.

Given that, the probability is around 50-60%, with moderate confidence. So Beta(6,4) would be mean 0.6, with alpha+beta=10. That seems reasonable.

④ Model completion:

<distribution>Beta(6.0, 4.0)</distribution>
*0.60*

output-first (commit, then reason) — P(Yes) = 0.500, Brier 0.250

② Model reasoning (<think>) under this prompt format:

Okay, let's tackle this. The question is whether Apple will launch an iPhone SE on February 19. The resolution criteria say that if they announce it during the event on that date, it's a yes. Otherwise, no. The event is scheduled for February 19, as per Tim Cook's tweet.

Looking at the news articles, there's a mix of information. Some articles suggest the iPhone SE 4 is expected to be announced soon. For example, article [2] from Apple Insider says the iPhone SE 4 is expected by February 21, with a smaller announcement on Wednesday (which would be February 12, but that's before the 19th). Article [7] mentions an expected announcement on February 11, but that's already passed. However, article [1] from MacRumors says the launch is expected in early 2025, which is vague but could include February.

Article [6] from Apple Insider mentions a possible announcement as soon as the week of February 14, which is close to the 19th. Article [10] from 9to5Mac suggests March as the likeliest date, but that's after February 19. However, other articles like [4] and [6] mention possible launches as early as this week (February 11-14) or later in spring.

There's also article [8] from MacWorld that says February is slow for Apple, with no mention of an iPhone SE launch on the 19th, and Mark Gurman suggests holding it for a March event. But the resolution criteria say that if the event on February 19 happens and they announce it then, it's a yes. However, the user's question is about whether the event on February 19 will include the iPhone SE launch.

But the key point is that the event is scheduled for February 19, and the question is whether during that event, Apple will announce the iPhone SE. The news articles are conflicting. Some suggest the SE 4 is coming soon, but others point to March. However, the latest articles (from February 13, 12, 11) are more recent and some mention the possibility of an announcement in the next week (February 19 is within that timeframe).

But article [7] says it's expected to be announced via press release, not a big event. However, the resolution criteria mention that the event is scheduled, and if they announce it during that event, it's a yes. If they don't announce it during the event, it's a no.

So, considering that some articles suggest the SE 4 is expected to be announced in early February (like the week of Feb 14), but others say March. However, the event is on Feb 19. If the announcement is scheduled for the event, then yes. But if the latest info (like article [8] from MacWorld) says that February is slow and no mention of SE on 19th, but other articles are more optimistic.

Also, the user's current date is February 14, so the event is in 5 days. The articles from February 13 and 12 are very recent. Article [1] says the launch is expected in early 2025, which is vague. Article [2] says the iPhone SE 4 is expected by Feb 21, which is after the 19th. But the event is on the 19th. If they announce it on the 19th, then yes. But if the announcement is on the 21st, then no.

But the resolution criteria say that if the event on Feb 19 happens and they announce it during that event, it's a yes. If the event doesn't happen by the deadline (Feb 19 11:59 PM ET), it's a no. However, the event is scheduled, so it's happening. The question is whether during that event, they announce the SE.

Given that some articles suggest the SE 4 is expected to be announced soon, but others are more cautious. The fact that Apple has scheduled an event on Feb 19, and the SE 4 is a major product, it's possible they might announce it then. However, other reports (like article [8]) suggest that the SE might be held for a March event.

But the user's question is about the event on Feb 19. If the event is happening, and if they announce the SE during that event, then yes. The probability would depend on whether the latest info points to the SE being announced on that date.

Considering the conflicting reports, but the fact that some articles (like [2] and [6]) mention the SE 4 being announced in the next week (which includes Feb 19), and the event is scheduled, I think there's a moderate chance. However, the article [8] from MacWorld says that February is slow and no mention of SE on 19th, which might lower the probability.

But since the event is scheduled, and Apple might use it to announce the SE, even if some reports are conflicting, I think the probability is around 0.5. But maybe a bit higher because the event is happening and the SE is expected soon. However, the article [7] says it's a press release, not a big event. But the resolution criteria say that if they announce it during the event, it's a yes. So if the event is the one on Feb 19, and they announce it there, then yes.

But given that some sources say the SE is expected in March, and others in early Feb, but the event is on Feb 19, I think the probability is around 0.4 to 0.5. But since the event is scheduled, and the SE is expected soon, maybe 0.55. However, the article [8] from MacWorld, which is from Feb 3, says that February is slow and no mention of SE on 19th. But other articles are more recent.

Considering the latest articles (from Feb 13, 12, 11) suggest the SE is expected soon, but not necessarily on the 19th. However, the event is on the 19th, and Apple might use that event to announce it. So maybe 0.55. But I'm not sure. Alternatively, since some articles suggest March, and others suggest Feb 21, which is after the 19th, the probability might be lower.

I think the best estimate is around 0.45 to 0.5. Given the conflicting info, I'll go with 0.5.

④ Model completion:

<probability>0.50</probability>

Set-matched comparison (added 2026-06-18)

Parse failure is informative abstention: a model that declines 20% of questions is scored only on the (easier) 80% it answered, so a parse-conditioned Brier flatters it. The project-wide fix (see Overview → Measurement standards) is to compare every method on the matched subset — the questions all methods in the group parsed — and to show parse rate beside the score. Full-coverage references (o1, ReMax, Polymarket) are scored on the same subset. Tooling: 4-regression-forecasting/forecasting_matched.py.

The 4B reward comparison (log vs Brier vs Beta-format RL), matched n=1005:

4B method	parse %	Brier (own / matched)	ECE	Acc	extreme
log-reward LoRA	0.997	0.241 / 0.232	0.129	0.651	0.044
Brier-reward LoRA	0.993	0.256 / 0.236	0.122	0.650	0.056
Beta-format RL	0.802	0.213 / 0.213	0.092	0.693	0.008
o1 (ref)	1.000	0.202 / 0.207	0.101	0.691	0.092
ReMax (ref)	1.000	0.191 / 0.192	0.066	0.710	0.074
Polymarket (ref)	1.000	0.151 / 0.154	0.050	0.779	0.272

On the matched subset Beta-format RL still leads the 4B field (0.213 vs log 0.232) — that win is real, not an artifact. But log/Brier improve when restricted to the questions Beta answered (0.241→0.232, 0.256→0.236), confirming those 1005 questions are easier; and Beta carries a 20-point lower parse rate. The honest reading: Beta is better per answered question while answering fewer of them.

The base-model format/scale comparison (14B/32B point vs Beta), matched n=964:

base method	parse %	Brier (own / matched)	ECE	Acc	extreme
14B point probability	0.990	0.348 / 0.282	0.186	0.632	0.240
14B Beta, zero-shot	0.809	0.203 / 0.200	0.089	0.682	0.024
32B Beta, zero-shot	0.912	0.204 / 0.202	0.105	0.691	0.023

The point format's Brier falls hard on the matched subset (0.348→0.282): the questions Beta abstains on are exactly where the point format flails. The Beta win survives matching (0.200 vs 0.282) but is ~40% smaller than the parse-conditioned gap implied.

Example rollouts — per method (click to expand)

For each method, a confident-correct rollout and a confident-wrong one (the overconfident miss is the failure a proper score punishes). Each shows the full prompt, reasoning, gold resolution, and completion.

Base Qwen3-4B (point probability) — parse rate 1.000, Brier 0.486

✓ Good — confident & correct (Brier 0.02)

① Prompt (superforecaster instruction + question + news + answer-format ask):

You are an expert superforecaster, familiar with Structured Analytic Techniques as well as Superforecasting by Philip Tetlock and related work. Predict the probability that the following question will be resolved as true/yes. You MUST give a probability estimate between 0 and 1 UNDER ALL CIRCUMSTANCES.

Question:
Will the highest temperature in NYC be between 36-37°F on March 3?

Question Background:

Resolution Criteria:
This market will resolve to the temperature range that contains the highest temperature recorded at the LaGuardia Airport Station in degrees Fahrenheit on March 3, 2025.

The resolution source for this market will be information from Wunderground, specifically the highest temperature recorded for all times on this day by the Forecast for the LaGuardia Airport Station once information is finalized, available here: https://www.wunderground.com/history/daily/us/ny/new-york-city/KLGA.

This market can not resolve to "Yes" until all data for this date has been finalized.

The resolution source for this market measures temperatures to whole degrees Fahrenheit (eg, 21°F). Thus, this is the level of precision that will be used when resolving the market.

Any revisions to temperatures recorded after data is finalized for this market's timeframe will not be considered for this market's resolution.

Today's date: 2025-03-01 16:00:05+00:00
Question close date: 2025-03-03 00:00:00+00:00

We have retrieved the following information for this question:
<News>
---
ARTICLES
[1] Split Weekend In NYC: Before And After The Front (published on 2025-02-28T12:00:00.000Z by weather.com)
Summary: A cold front will hit NYC, making Saturday warmer but gusty, and Sunday chilly, with the weekend split into two different forecasts.

[2] N.Y. weather: Forecast for first half of March reveals shift from ... (published on 2025-02-20T12:00:00.000Z by silive.com)
Summary: New York is forecast to have below-average temperatures from March 1-14, 2025, with a 55-60% likelihood, and average temperatures in the area typically range from 38 to 42 degrees Fahrenheit during this period.

[3] New York's spring 2025 forecast: Here's what Old Farmer's Almanac ... (published on 2025-02-12T12:00:00.000Z by lohud.com)
Summary: A warm, wet spring is predicted for the New York City metro area, including Long Island. The Old Farmer's Almanac predicts warmer-than-normal temperatures for most of the country, with above average rainfall. The National Weather Service's Climate Prediction Center expects average or above average temperatures from March to May, with about half the nation getting average precipitation. AccuWeather predicts some cold periods in New York through February and March, but a more extended forecast is not available.

[4] Farmers Almanac spring 2025 weather forecast for New York (published on 2025-02-10T12:00:00.000Z by pressconnects.com)
Summary: The National Weather Service and Old Farmer's Almanac predict above-average temperatures and precipitation for the Southern Tier region this spring, while AccuWeather forecasts cooler temperatures and a chance for a late frost, with a higher chance of late-season snow, and slightly lower than average temperatures from March to May.

[5] Early spring or more winter? Meteorologist's 2025 prediction for NYC (published on 2025-02-02T12:00:00.000Z by fox5ny.com)
Summary: FOX 5 NY Meteorologist Nick Gregory expects 7-12" more snow for NYC and Long Island, with warmer than average spring temperatures, but near average precipitation, and the first 60-degree day usually lands on April 10; Punxsutawney Phil's prediction accuracy is 30% over the last 10 years, and the Climate Prediction Center forecasts average temperatures for NYC in the week after Groundhog Day.
----
</News>

Think about your answer and output your final prediction (a number between 0 and 1) with an asterisk at the beginning and end of the decimal (Ex: *<probability>*).

② Model reasoning (<think>):

Okay, let's tackle this. The question is whether the highest temperature in NYC on March 3, 2025, will be between 36-37°F. The resolution is based on the LaGuardia Airport's data, and the answer has to be a probability between 0 and 1.

First, I need to look at the context provided. The user has given several news articles from around February 2025. Let me go through each of them to see what they say about the weather in NYC for March.

Article 1 from weather.com mentions a cold front hitting NYC, making Saturday (March 1) warmer but gusty, and Sunday (March 2) chilly. But the question is about March 3, so maybe the weekend's weather is a clue. However, the article is from February 28, so it's a forecast for the weekend of March 1-2. The cold front might affect the temperatures, but the exact date is March 3.

Article 2 from silive.com says that March 1-14 will have below-average temperatures with 55-60% likelihood. The average temperatures in NYC during this period are usually 38-42°F. So if it's below average, maybe the temperatures would be lower than that range. But the question is about March 3, which is in the middle of that period. If the average is 38-42, and it's below average, maybe the temperatures would be lower, but how much lower?

Article 3 from lohud.com says a warm, wet spring is predicted, with warmer-than-normal temps for most of the country. However, the NWS Climate Prediction Center expects average or above average temps from March to May. So there's a bit of a contradiction here. The Old Farmer's Almanac says warmer, but the NWS says average or above. But the article also mentions that AccuWeather forecasts some cold periods in NYC through February and March, but no extended forecast. So maybe there's a mix of predictions.

Article 4 from pressconnects.com says that the National Weather Service and Old Farmer's Almanac predict above-average temps and precipitation for the Southern Tier, but AccuWeather forecasts cooler temps and a chance for late frost. However, the Southern Tier is a different region, so maybe NYC is not included here. The article also mentions that from March to May, there's a higher chance of lower temps. But again, this is conflicting with the previous article.

Article 5 from fox5ny.com says that the meteorologist expects 7-12" more snow for NYC and Long Island, with warmer than average spring temps. But the first 60-degree day is expected on April 10. The Climate Prediction Center forecasts average temps for NYC in the week after Groundhog Day (which is February 2nd). So March is after that, so maybe the average temps would be around normal. But the first 60-degree day is in April, so March might still be colder.

Putting this all together, there's a mix of predictions. Some sources say below average temps for March 1-14 (article 2), others say average or above (article 3). The Old Farmer's Almanac says warmer, but the NWS says average. The AccuWeather says some cold periods, but not sure about March 3 specifically.

The key is that the average temperature in March for NYC is typically 38-42°F. If the forecast is for below average, then March 3 might be lower. But if the average is 38-42, and the question is about 36-37, which is below average, then maybe the probability is low. However, the articles are conflicting. Article 2 says 55-60% chance of below average temps, which would mean that March 3 might be lower than average. But article 3 says that the NWS expects average or above. So which is more reliable?

Tetlock's work suggests that superforecasters use structured analytic techniques, considering base rates and reference classes. The base rate for March temperatures in NYC is 38-42°F. The question is about 36-37, which is 2-3 degrees below average. The articles are conflicting, but article 2 is from silive.com, which is a local news source, and it's more specific about March 1-14. The article says 55-60% chance of below average temps. If that's the case, then March 3 is in that period, so the chance of below average is high. But the other articles suggest that maybe it's average or above. However, the article from silive.com is more specific and mentions the likelihood of below average temps.

But also, the article from weather.com mentions a cold front on the weekend (March 1-2), which might make March 3 colder. However, the cold front is on the weekend, so March 3 is after that. If the cold front made the weekend (March 1-2) colder, then March 3 might be milder. But the article from silive.com says that March 1-14 is below average. So if that's the case, then March 3 would be in that period, so the temperature would be below average.

But the average is 38-42, so 36-37 is 2-3 degrees below. If the average is 38-42, then the chance of getting 36-37 would be low. However, the articles are conflicting. The Old Farmer's Almanac says warmer, but the NWS says average. But the silive.com article is more specific and says 55-60% chance of below average.

So, considering that, the probability that the temperature is between 36-37°F would be low. But how low? If the average is 38-42, and the chance of below average is 55-60%, then the chance of being 36-37 would be part of that below average. But how much of the below average is that? If the average is 38-42, then 36-37 is 2-3 degrees below. If the distribution is normal, then the probability of being 2-3 degrees below average would be around 10-15% (since in a normal distribution, about 68% is within 1 SD, 95% within 2 SD, etc. But if the average is 38-42, then 36-37 is 2-3 degrees below. Assuming that the standard deviation is around 2-3 degrees, then 36-37 would be about 1-1.5 SD below average. The probability of that would be around 10-15% (since 95% is within 2 SD, so 2.5% in each tail). But this is all speculative.

Alternatively, if the average is 38-42, and the chance of below average is 55-60%, then the chance of being 36-37 would be part of that. But how much? If the below average is 55-60%, and the 36-37 is a subset of that, maybe the probability is around 10-15%? But I'm not sure.

Alternatively, maybe the temperature on March 3 is more likely to be in the 38-42 range, but the articles are conflicting. However, the silive.com article is more specific and says that March 1-14 is below average with 55-60% chance. So if that's the case, then the temperature would be below average, which is 38-42°F. But the question is about 36-37, which is lower than that. So maybe the probability is low.

But the problem is that the articles are conflicting. The Old Farmer's Almanac says warmer, but the NWS says average. However, the silive.com article is from a local source and is more specific. Also, the article from weather.com mentions a cold front on the weekend, which might make March 3 colder.

But the resolution is based on the actual data from Wunderground. Since the user is asking for a probability, and the articles are conflicting, but the silive.com article is more specific about the March 1-14 period, which includes March 3, and says 55-60% chance of below average temps. So if that's the case, then the temperature would be below average, which is 38-42°F. But the question is about 36-37, which is lower than that. So the probability would be low.

But how low? If the average is 38-42, and the chance of below average is 55-60%, then the chance of being 36-37 would be part of that. But how much? If the distribution is normal, then the probability of being 2-3 degrees below average would be around 10-15%. However, if the temperatures are not normally distributed, maybe it's different.

Alternatively, maybe the average is 38-42, and the chance of being 36-37 is low. So the probability would be around 0.10 (10%) or 0.15 (15%). But I need to consider the reference class. The reference class

③ Gold resolution: No (0) (reference probs — Polymarket 0.32, o1 0.20)

④ Model completion (answer after reasoning):

[no </think> — reasoning ran to the token limit]

Predicted P(Yes) = 0.150; Brier (this question) = 0.022 — the proper score rewards committing here

✗ Overconfident miss — confident on the wrong side (Brier 1.00)

① Prompt (superforecaster instruction + question + news + answer-format ask):

You are an expert superforecaster, familiar with Structured Analytic Techniques as well as Superforecasting by Philip Tetlock and related work. Predict the probability that the following question will be resolved as true/yes. You MUST give a probability estimate between 0 and 1 UNDER ALL CIRCUMSTANCES.

Question:
Will Apple launch an iPhone SE on February 19?

Question Background:


Resolution Criteria:
Apple has scheduled a launch event for February 19: https://x.com/tim_cook/status/1890068457825394918

This market will resolve to "Yes" if Apple announces a new iPhone SE product during this event. Otherwise, this market will resolve to "No".

If multiple products are released, each relevant market may resolve to "Yes".

This market may not resolve to "No" until the February 19 launch event is concluded.

If this event does not occur by February 19, 2025, 11:59 PM ET, this market will resolve to "No".

The primary resolution source for this market will be an official feed of the event and information from Apple, however a consensus of credible reporting will also be used.

Today's date: 2025-02-14 16:00:04+00:00
Question close date: 2025-02-19 00:00:00+00:00



We have retrieved the following information for this question:
<News>
---
ARTICLES
[1] iPhone SE 4 Rumors: Price, Features, Release Date and More (published on 2025-02-13T12:00:00.000Z by macrumors.com)
Summary: Apple is developing a new iPhone SE with a larger 6.06-inch display, all-display design, and Face ID, replacing Touch ID. It's expected to feature a USB-C port, single-lens camera, and A18 chip, supporting Apple Intelligence. The launch is expected in early 2025, with a possible price starting at $429.

[2] iPhone SE 4 & Apple Vision Pro announcements are coming in the next week (published on 2025-02-12T00:00:00.000Z by appleinsider.com)
Summary: Apple is preparing announcements, with the iPhone SE 4 expected by February 21. A smaller announcement is planned for Wednesday, and Apple Vision Pro briefings are set for Friday, possibly related to an operating system or app update. The M4 MacBook Air is expected within weeks, around March. Product briefings for the iPhone SE 4 will occur before its announcement, which is expected soon.

[3] iPhone SE 4 coming soon — 7 biggest rumored upgrades (published on 2025-02-11T12:00:00.000Z by tomsguide.com)
Summary: Apple is expected to launch a new midrange phone, the iPhone SE 4, which will be significantly different from the current model, with rumors suggesting a launch as early as this week or later this spring, featuring a new design, more powerful chipset, Apple Intelligence support, camera upgrade, OLED display, and USB-C port, with a possible price hike to under $500.

[4] I revisited the iPhone SE in 2025 - and the SE 4 may be an even bigger hit (published on 2025-02-11T00:00:00.000Z by zdnet.com)
Summary: Apple is expected to launch a new iPhone SE this week, replacing the current model that launched almost three years ago, with significant upgrades including a 6.1-inch OLED display, Face ID, 48MP main camera, A18 chip, and USB-C port, pricing is expected to be competitive.

[5] iPhone SE 4 Date/Time? (published on 2025-02-10T00:00:00.000Z by forums.macrumors.com)
Summary: Apple users are speculating about the release of the iPhone SE 4, with some expecting it to be announced this week. However, there was no official announcement, and previous SE releases were announced in March. Some think it might be announced on Tuesday or Wednesday, with preorders starting on Friday, while others believe it will be released in March alongside other Apple products. No one knows for sure when it will be released, but the lack of an announcement on Monday has sparked speculation.

[6] iPhone SE 4 rumors: Price, screen, cameras, release date (published on 2025-02-07T12:00:00.000Z by appleinsider.com)
Summary: Apple may announce the fourth-generation iPhone SE as soon as the week of February 14, 2025, with the first units expected to reach customers, and it's expected to feature a design derived from the iPhone 14, with a 6.1-inch OLED display, Face ID, and a 48 MP rear camera.

[7] Long Wait Over For Budget iPhone Buyers As New SE 4 Ready For Release - 9meters (published on 2025-02-07T00:00:00.000Z by 9meters.com)
Summary: The iPhone SE 4 is expected to be announced on February 11, 2025, with pre-sales starting immediately after or on February 14th, and the full release between February 21 and 28th, featuring flagship features at a lower cost, and will be announced through a press release, not a big event.

[8] Here's everything we expect from Apple in February 2025 (published on 2025-02-03T00:00:00.000Z by macworld.com)
Summary: February is expected to be slow for Apple releases, with a possible "direct to the website" launch of the M4 MacBook Air, but no mention of an iPhone SE launch on February 19, and Bloomberg's Mark Gurman suggests the iPhone SE 4 might be held for a bigger event in March.

[9] The MacRumors Show: Latest iPhone SE 4 Rumors (published on 2025-01-31T12:00:00.000Z by macrumors.com)
Summary: The iPhone SE 4 is expected to feature an iPhone 14-style design, 6.1-inch OLED display, Face ID, and USB-C, with a launch expected before iOS 18.4, around March or April, at a price of about $470.

[10] iPhone SE 4 release date: When to expect Apple’s powerful upgrade - 9to5Mac (published on 2025-01-29T00:00:00.000Z by 9to5mac.com)
Summary: Apple is expected to launch the iPhone SE 4 in early 2025, with March being the likeliest date, based on reporting from the supply chain regarding the production timeline and past iPhone SE launch dates (March 18, 2022, April 24, 2020, and March 24, 2017).
----
</News>

Think about your answer and output your final prediction (a number between 0 and 1) with an asterisk at the beginning and end of the decimal (Ex: *<probability>*).

② Model reasoning (<think>):

Okay, let's tackle this. The question is whether Apple will launch the iPhone SE on February 19, 2025. The user provided a bunch of news articles from various sources, so I need to go through them and see what the consensus is.

First, the background says that Apple has a scheduled launch event on February 19. The market resolves to "Yes" if they announce the new iPhone SE during that event. If not, it's "No". Also, if the event doesn't happen by February 19, it's "No". The resolution is based on official feeds and credible reporting.

Looking at the articles:

Article 1 (macrumors.com) from Feb 13 says Apple is developing a new iPhone SE with a 6.06-inch display, Face ID, USB-C, A18 chip, etc., expected in early 2025. The price is $429. So they're developing it, but the exact date isn't given here.

Article 2 (appleinsider.com) from Feb 12 mentions that the iPhone SE 4 is expected by Feb 21, with a smaller announcement planned for Wednesday (Feb 13?), and Vision Pro on Friday. The M4 MacBook Air is expected in March. So the SE 4 is expected by Feb 21, but the event is on Feb 19. So maybe the announcement is on the 19th?

Article 3 (tomsguide.com) from Feb 11 says the SE 4 is expected to be launched this week or later this spring. The article mentions a possible price hike to under $500. The date here is Feb 11, so "this week" would be around Feb 13-17, but the event is on the 19th. So maybe the announcement is on the 19th?

Article 4 (zdnet.com) from Feb 11 says the SE 4 is expected this week, with significant upgrades. The article mentions a 6.1-inch OLED display, Face ID, etc. Again, the date is Feb 11, so "this week" would be around the 13th-17th. But the event is on the 19th.

Article 5 (forums.macrumors.com) from Feb 10 says some expect it to be announced this week, but no official announcement yet. Previous SEs were in March. Some think it might be on Tuesday or Wednesday (Feb 12-13?), but others think March. However, the event is on the 19th, so maybe the announcement is on the 19th.

Article 6 (appleinsider.com) from Feb 7 says the SE 4 might be announced as early as the week of Feb 14. The first units would be available, and the design is similar to the iPhone 14. So the week of Feb 14 would be around the 14th-20th. The event is on the 19th, so that's within that window.

Article 7 (9meters.com) from Feb 7 says the SE 4 is expected on Feb 11, with pre-sales starting on the 14th. The full release between Feb 21-28. But the event is on the 19th, so maybe the announcement is on the 19th?

Article 8 (macworld.com) from Feb 3 says February is slow for Apple, and no mention of the SE launch on the 19th. However, Mark Gurman suggests the SE 4 might be held for a bigger event in March. But this contradicts some other articles. However, the event is on the 19th, so if the SE is not announced then, maybe it's pushed to March.

Article 9 (macrumors.com) from Jan 31 says the SE 4 is expected to be launched before iOS 18.4, around March or April. So this suggests a later date, but the event is on the 19th.

Article 10 (9to5mac.com) from Jan 29 says the SE 4 is expected in early 2025, with March as the likeliest date, based on production timelines and past dates (March 2022, etc.). So again, March is the likely date, but the event is on the 19th.

Now, the key is to see if the articles are pointing to the 19th as the date. Some articles (like 2, 6, 7) mention the SE 4 being announced in the week of Feb 14, which would include the 19th. However, article 8 says that February is slow and the SE might be pushed to March. But the event is on the 19th, so if Apple is not announcing it then, it's possible they might have a different plan.

But the user's background says that Apple has scheduled a launch event on the 19th. So the event is definitely happening. The question is whether the SE is announced during that event. The articles are conflicting. Some say the SE is expected by Feb 21 (article 2), others say the week of Feb 14 (article 6), and some suggest March (articles 8, 9, 10). However, the event is on the 19th, so if the SE is announced during that event, it's a "Yes". If not, then "No".

But the problem is that the articles are conflicting. For example, article 2 says the SE is expected by Feb 21, but the event is on the 19th. So maybe the announcement is on the 19th. However, article 8 says that the SE might be held for a bigger event in March. But the event is on the 19th, so if Apple is not announcing it then, it's possible they might not. However, the event is scheduled, so they might use that as the date.

Also, the user's background says that the event is scheduled, so the market is based on whether Apple announces the SE during that event. The articles are mostly pointing to the SE being announced in early February, but some are conflicting. However, the most recent articles (from Feb 13, 12, 11, etc.) are suggesting that the SE is expected to be announced in the week of Feb 14, which would include the 19th. However, article 8 says that February is slow and the SE might be pushed to March. But the event is on the 19th, so if Apple is not announcing it then, it's possible they might not. However, the event is scheduled, so they might use that as the date.

But the problem is that the articles are conflicting. Some say the SE is expected by Feb 21, others by the week of Feb 14. The event is on the 19th, so if the SE is announced during that event, it's a "Yes". If not, then "No".

But given that the event is scheduled, and Apple is likely to use that as the date for the SE announcement, even if some articles suggest March, the most recent articles (from Feb 13, 12, 11) are pointing to the SE being announced in early February. However, article 8 says that February is slow and the SE might be pushed to March. But the event is on the 19th, so if Apple is not announcing it then, it's possible they might not. However, the event is scheduled, so they might use that as the date.

But the key is that the event is scheduled, and the market is based on whether Apple announces the SE during that event. The articles are conflicting, but the most recent ones (from Feb 13, 12, 11) suggest that the SE is expected to be announced in early February, which would include the 19th. However, article 8 says that the SE might be pushed to March. But the event is on the 19th, so if Apple is not announcing it then, it's possible they might not. However, the event is scheduled, so they might use that as the date.

But given that the event is scheduled, and the articles are pointing to the SE being announced in early February, I think the probability is higher. However, there's a conflict with article 8. But since the event is scheduled, and Apple is likely to use that as the date, the probability is higher. However, the articles are conflicting. So I need to weigh the evidence.

The most recent articles (from Feb 13, 12, 11) suggest that the SE is expected to be announced in early February, which would include the 19th. However, article 8 says that February is slow and the SE might be pushed to March. But the event is on the 19th, so if Apple is not announcing it then, it's possible they might not. However, the event is scheduled, so they might use that as the date.

But the problem is that the articles are conflicting. However, the user's background says that Apple has scheduled the event on the 19th. So the event is definitely happening. The question is whether the SE is announced during that event. The articles are pointing to the SE being announced in early February, which would include the 19th. However, some articles suggest March. But the event is on the 19th, so if Apple is not announcing

③ Gold resolution: No (0) (reference probs — Polymarket 0.92, o1 0.30)

④ Model completion (answer after reasoning):

[no </think> — reasoning ran to the token limit]

Predicted P(Yes) = 1.000; Brier (this question) = 1.000 — exactly what the log/Brier reward penalizes

4B + log-reward LoRA (s300) — parse rate 0.997, Brier 0.241

✓ Good — confident & correct (Brier 0.00)

① Prompt (superforecaster instruction + question + news + answer-format ask):

Question:
Will the highest temperature in NYC be between 48-49°F on February 28?

Question Background:

Resolution Criteria:
This market will resolve to the temperature range that contains the highest temperature recorded at the LaGuardia Airport Station in degrees Fahrenheit on February 28, 2025.

This market can not resolve to "Yes" until all data for this date has been finalized.

The resolution source for this market measures temperatures to whole degrees Fahrenheit (eg, 21°F). Thus, this is the level of precision that will be used when resolving the market.

Any revisions to temperatures recorded after data is finalized for this market's timeframe will not be considered for this market's resolution.

Today's date: 2025-02-26 16:00:05+00:00
Question close date: 2025-02-28 00:00:00+00:00

We have retrieved the following information for this question:
<News>
---
ARTICLES
[1] Weather in New York City in February 2025 (New York) - detailed Weather Forecast for a month (published on 2025-02-23T00:00:00.000Z by world-weather.info)
Summary: New York City's February 2025 weather forecast: highs range from +25° to +46°, with no temperature above 46° listed, including Feb 28's high of +30°.

[2] National Weather Service (published on 2025-02-13T00:00:00.000Z by forecast.weather.gov)
Summary: Today Rain, high near 49. Tonight low around 29. Friday high near 36. Saturday high near 39. Sunday high near 52. Monday high near 30. Tuesday high near 28. Wednesday high near 31.

[3] Snow and ice are about to hit the northern U.S. Here’s where. (published on 2025-02-08T00:00:00.000Z by washingtonpost.com)
Summary: A pair of winter storms will hit the northern US, with the first bringing 10 inches of snow to New England through Sunday, and potentially impacting Super Bowl travel. New York City will see 2-4 inches of snow and minor ice buildup, with the heaviest snow from 7 p.m. to midnight, and temperatures in the mid-30s.

[4] Winter Storm Garnett: A foot of snow expected in the Northeast U.S. on Sunday (published on 2025-02-06T00:00:00.000Z by severe-weather.eu)
Summary: A strong Polar Vortex is affecting the Northern Hemisphere, with a significant winter storm, Garnett, forecast to hit the Northeast US on Sunday, bringing a foot of snow and freezing rain. The vortex's southern lobe is located over North America, triggering cold outbreaks and winter storms. Temperatures are expected to remain cold, with no indication of warming to 48-49°F in the near future, including February 28.

[5] Early spring or more winter? Meteorologist’s 2025 prediction for NYC (published on 2025-01-31T00:00:00.000Z by fox5ny.com)
Summary: FOX 5 NY Meteorologist Nick Gregory expects 7-12" more snow for NYC and Long Island, with warmer than average spring temperatures, but "probably near average" precipitation, and the first 60-degree day usually lands on April 10, with February bringing a weather pattern change with a westward shift in the jet stream allowing warmer temperatures to spread across the eastern half of the country.
----
</News>

Think about your answer and output your final prediction (a number between 0 and 1) with an asterisk at the beginning and end of the decimal (Ex: *<probability>*).

② Model reasoning (<think>):

Okay, let's tackle this probability question. The user wants to know the chance that the highest temperature in NYC on February 28, 2025, will be between 48-49°F. The resolution is based on data from Wunderground, and the temperature is recorded to the nearest whole degree.

First, I need to look at the provided news articles to gather relevant information. Let's go through each one.

Article 1 from world-weather.info says the February 2025 forecast has highs ranging from 25° to 46°, with Feb 28's high at 30°. That's a bit low, but it's a forecast. However, forecasts can be inaccurate, so I need to check other sources.

Article 2 is from the National Weather Service. It mentions that on February 28, the high is near 36°F. Wait, that's conflicting with the first article. But the NWS is a more official source, so maybe that's more reliable? But the date here is February 13, so it's a forecast for the week ahead. The high on Feb 28 is near 36°F. But the user is asking about the actual recorded temperature, not the forecast. However, the actual data isn't available yet, so we have to rely on these forecasts.

Article 3 and 4 talk about winter storms affecting the northern US, including NYC. Article 3 mentions 2-4 inches of snow and temperatures in the mid-30s. Article 4 says a strong polar vortex with cold temperatures, no warming to 48-49°F. So these articles suggest that temperatures would be cold, maybe in the 30s or lower.

Article 5 is a meteorologist's prediction expecting more snow but mentions that the first 60-degree day is around April 10. So February is likely still cold. The article also says February brings a weather pattern change, but the jet stream shift allows warmer temps in the east. However, the specific date is Feb 28, which is still early in the year. The forecast in article 2 says high near 36°F on Feb 28, but that's a forecast. However, the actual data isn't available yet, so we have to rely on the best available information.

The question is about the actual recorded temperature. The problem is that the articles are all forecasts, and the actual data isn't available yet. However, the user is asking for the probability based on the given information. The key is to determine if the temperature is likely to be in the 48-49°F range.

Looking at the forecasts, the highest high mentioned is 36°F in article 2, and the world-weather.info says 30°F. The other articles suggest cold temperatures, mid-30s. The only mention of higher temps is in article 1, which says 46°F as the upper limit for February. But that's a forecast, and the actual data might be different.

However, the user is asking for the probability that the actual temperature will be between 48-49°F. Given that all the sources indicate that the high is in the 30s or lower, and there's no mention of temperatures reaching 48-49°F, it's unlikely. The only possible source of higher temps is if the actual data shows a higher temperature, but the articles don't indicate that. The NWS forecast for Feb 28 is 36°F, which is lower than 48-49. The other articles also suggest cold weather. Therefore, the probability is low. The base rate for such a temperature in February in NYC is low, as February is typically colder. The reference class would be daily high temperatures in NYC in February, which are usually in the 20s to 40s. So 48-49 is way above the usual range. Therefore, the probability is very low, maybe around 5% or lower. But since the user wants a precise number, and considering that the forecasts are all below 48, I would estimate a low probability, perhaps 0.05 or 0.07.

③ Gold resolution: No (0) (reference probs — Polymarket 0.35, o1 0.04)

④ Model completion (answer after reasoning):

<probability>0.05</probability>

Predicted P(Yes) = 0.050; Brier (this question) = 0.003 — the proper score rewards committing here

✗ Overconfident miss — confident on the wrong side (Brier 0.56)

① Prompt (superforecaster instruction + question + news + answer-format ask):

You are an expert superforecaster, familiar with Structured Analytic Techniques as well as Superforecasting by Philip Tetlock and related work. Predict the probability that the following question will be resolved as true/yes. You MUST give a probability estimate between 0 and 1 UNDER ALL CIRCUMSTANCES.

Question:
Will the following market resolve to "Miami (OH)"? Miami (OH) vs. Western Michigan

Question Background:


Resolution Criteria:
In the upcoming CBB game, scheduled for February 15 at 2:30PM ET:
If the Miami (OH) win, the market will resolve to "Miami (OH)".
If the Western Michigan win, the market will resolve to "Western Michigan".
If the game is postponed, this market will remain open until the game has been completed.
If the game is canceled entirely, with no make-up game, this market will resolve 50-50.

Today's date: 2025-02-14 16:00:04+00:00
Question close date: 2025-02-22 00:00:00+00:00



We have retrieved the following information for this question:
<News>
---
ARTICLES
[1] Miami (OH) at Western Michigan odds, tips and betting trends (published on 2025-02-12T12:00:00.000Z by sportsbookwire.usatoday.com)
Summary: Miami (OH) is a 6.5-point favorite over Western Michigan, with a predicted score of 79-71 in favor of Miami (OH), and has a 74.8% implied win probability based on the moneyline odds.

[2] Bronco Offense Explodes in 8-3 Victory Over Miami (published on 2025-01-24T12:00:00.000Z by wmubroncos.com)
Summary: Western Michigan's hockey team beat Miami 8-3, improving to 16-4-1 overall, but this article does not mention an upcoming CBB game between Miami (OH) and Western Michigan, only a hockey game.

[3] Western Michigan hockey scores EIGHT goals on Friday night (published on 2025-01-24T12:00:00.000Z by fox17online.com)
Summary: Western Michigan hockey, ranked No. 3, scored eight goals against Miami (OH) on Friday night, defeating them.
----
</News>

Think about your answer and output your final prediction (a number between 0 and 1) with an asterisk at the beginning and end of the decimal (Ex: *<probability>*).

② Model reasoning (<think>):

Okay, let's tackle this probability question. The user wants to know the chance that the CBB game between Miami (OH) and Western Michigan will resolve to "Miami (OH)" on February 15, 2025. The current date is February 14, 2025, so the game is just a day away.

First, I need to look at the relevant information provided. The news articles mention that Miami (OH) is a 6.5-point favorite over Western Michigan, with a predicted score of 79-71 in favor of Miami. The implied win probability for Miami is 74.8% based on the moneyline odds. That's a strong indicator that Miami is expected to win.

But wait, there's also a hockey game mentioned in articles 2 and 3. However, the question is about the CBB (College Basketball) game, not hockey. The hockey results don't directly affect the basketball game's outcome, so I can probably ignore those.

The resolution criteria say that if the game is postponed, the market remains open until it's completed. But since the game is scheduled for February 15, and the close date is February 22, there's a possibility of postponement, but the user is asking for the probability that the market resolves to Miami (OH). If the game is postponed, the market would stay open, but the question is about the probability of the game's outcome, not the market's resolution in case of postponement.

The key here is the implied probability from the sportsbook. A 74.8% chance based on moneyline odds is a strong indicator. However, I should consider if there are any other factors. The articles don't mention any recent developments or injuries that could affect the game. The only other info is the hockey game, which is unrelated.

Base rates and reference classes: In college basketball, favorites usually have a higher chance of winning, especially if they're a 6.5-point favorite. The 74.8% implied probability aligns with that. Also, the predicted score of 79-71 suggests a clear win for Miami.

I should also check if there's any reason to doubt the sportsbook's odds. But without any conflicting information, the 74.8% seems reliable. The market is likely to resolve as Miami (OH) wins, so the probability should be around 0.75. However, since the question is about the market resolving to "Miami (OH)", and the implied probability is 74.8%, which is close to 75%, I can round it to 0.75. But maybe the actual probability is slightly lower due to possible variables, but given the data, 0.75 is a reasonable estimate.

③ Gold resolution: No (0) (reference probs — Polymarket 0.51, o1 0.75)

④ Model completion (answer after reasoning):

*<probability>0.75</probability>*

Predicted P(Yes) = 0.750; Brier (this question) = 0.562 — exactly what the log/Brier reward penalizes

4B + Brier-reward LoRA (s300) — parse rate 0.993, Brier 0.255

✓ Good — confident & correct (Brier 0.02)

① Prompt (superforecaster instruction + question + news + answer-format ask):

Question:
Will the highest temperature in NYC be between 36-37°F on March 3?

Question Background:

Resolution Criteria:
This market will resolve to the temperature range that contains the highest temperature recorded at the LaGuardia Airport Station in degrees Fahrenheit on March 3, 2025.

This market can not resolve to "Yes" until all data for this date has been finalized.

The resolution source for this market measures temperatures to whole degrees Fahrenheit (eg, 21°F). Thus, this is the level of precision that will be used when resolving the market.

Any revisions to temperatures recorded after data is finalized for this market's timeframe will not be considered for this market's resolution.

Today's date: 2025-03-01 16:00:05+00:00
Question close date: 2025-03-03 00:00:00+00:00

Think about your answer and output your final prediction (a number between 0 and 1) with an asterisk at the beginning and end of the decimal (Ex: *<probability>*).

② Model reasoning (<think>):

Okay, let's tackle this question. The user wants to know the probability that the highest temperature in NYC on March 3, 2025, will be between 36-37°F. The resolution is based on data from Wunderground, and the market can't resolve until all data is finalized.

First, I need to consider the context provided. The articles mention that New York is forecasted to have below-average temperatures from March 1-14, with a 55-60% likelihood. The average temperatures typically range from 38-42°F during this period. However, there's also a mention of a cold front hitting NYC, making Saturday (March 2) warmer but gusty, and Sunday (March 3) chilly.

Wait, the question is about March 3. The article from weather.com says that the weekend is split, with Saturday (March 2) being warmer but gusty, and Sunday (March 3) chilly. So, if Sunday is chilly, maybe the temperature would be lower than average. But the average is usually 38-42°F. The question is about 36-37°F, which is below the average.

But the Old Farmer's Almanac and other sources predict a warm, wet spring, with above-average temperatures. However, the National Weather Service's Climate Prediction Center expects average or above average temperatures from March to May. But the article from silive.com says there's a 55-60% chance of below-average temps for March 1-14.

So there's a conflict here. The Climate Prediction Center says average or above, but the silive.com article says 55-60% chance of below average. Which one is more reliable? Maybe the silive.com is a local forecast, while the Climate Prediction Center is a broader national forecast.

Also, the article from fox5ny.com mentions that the first 60-degree day is usually April 10, suggesting that March might still be cold. The split weekend article says that March 3 (Sunday) is chilly.

But the question is about the highest temperature on March 3. If the day is chilly, maybe the high is lower. However, the average is 38-42, so 36-37 is below that. But the silive.com article says that the average is 38-42, so 36-37 would be below average. However, the chance of below average is 55-60%, but that's for the entire period.

But the specific date is March 3. The split weekend article says that Sunday (March 3) is chilly. So maybe the high would be lower. However, the average is 38-42, so 36-37 is a bit low. But the question is about the highest temperature. If the day is chilly, maybe the high is 36-37.

But how likely is that? The base rate for temperatures in March in NYC is usually 38-42. So 36-37 is below that. However, the articles mention that there's a possibility of colder temperatures, especially if a cold front is present.

But the problem is that the articles are from February, so they're predictions. The actual data for March 3, 2025, is not available yet. The user is asking for a probability based on the given articles.

The key is to assess the likelihood that the high would be 36-37. The silive.com article says that March 1-14 is likely to be below average, with 55-60% chance. If that's the case, then March 3 is part of that period. However, the split weekend article says that March 3 (Sunday) is chilly.

But the average is 38-42, so 36-37 is below average. However, the chance of below average is 55-60%, but that's for the entire period. The question is about a specific day.

But the split weekend article says that the weekend is split, with Saturday warmer and Sunday chilly. So maybe on March 3, the high is lower. However, the average is 38-42, so 36-37 is a bit low.

But how likely is that? If the average is 38-42, then 36-37 is 2-3 degrees below average. The question is whether that's within the range of possible outcomes.

The base rate for such a temperature in March would be low. However, the articles suggest that there's a possibility of colder temperatures. The silive.com article says that the chance of below average is 55-60%, but that's for the entire period.

But the split weekend article specifically mentions that March 3 is chilly. So maybe the high is lower. However, the actual data is not available yet.

Considering all this, the probability that the high is 36-37°F is low, but not zero. The articles suggest that there's a possibility of colder temperatures, but the average is 38-42. So maybe the probability is around 10-20%?

But I need to be careful. The base rate for March 3 in NYC is usually 38-42. The question is about 36-37, which is below average. The articles suggest that there's a chance of below average, but the exact probability for a specific day is not given.

The split weekend article says that Sunday is chilly, which might mean that the high is lower. However, the exact temperature is not specified.

Given that the average is 38-42, and the chance of below average is 55-60%, but the question is about a specific day. The probability might be around 10-15%?

Alternatively, maybe the chance is lower. If the average is 38-42, then 36-37 is 2-3 degrees below. The chance of that happening on a specific day might be lower than the overall period.

But the split weekend article says that March 3 is chilly, which might indicate that the high is lower. However, the exact temperature is not given.

Considering all the factors, I think the probability is around 10-15%. So the final answer would be 0.15.

③ Gold resolution: No (0) (reference probs — Polymarket 0.32, o1 0.20)

④ Model completion (answer after reasoning):

*<probability>0.15</probability>*

Predicted P(Yes) = 0.150; Brier (this question) = 0.022 — the proper score rewards committing here

✗ Overconfident miss — confident on the wrong side (Brier 1.00)

① Prompt (superforecaster instruction + question + news + answer-format ask):

You are an expert superforecaster, familiar with Structured Analytic Techniques as well as Superforecasting by Philip Tetlock and related work. Predict the probability that the following question will be resolved as true/yes. You MUST give a probability estimate between 0 and 1 UNDER ALL CIRCUMSTANCES.

Question:
Will Apple launch an iPhone SE on February 19?

Question Background:


Resolution Criteria:
Apple has scheduled a launch event for February 19: https://x.com/tim_cook/status/1890068457825394918

This market will resolve to "Yes" if Apple announces a new iPhone SE product during this event. Otherwise, this market will resolve to "No".

If multiple products are released, each relevant market may resolve to "Yes".

This market may not resolve to "No" until the February 19 launch event is concluded.

If this event does not occur by February 19, 2025, 11:59 PM ET, this market will resolve to "No".

The primary resolution source for this market will be an official feed of the event and information from Apple, however a consensus of credible reporting will also be used.

Today's date: 2025-02-14 16:00:04+00:00
Question close date: 2025-02-19 00:00:00+00:00



We have retrieved the following information for this question:
<News>
---
ARTICLES
[1] iPhone SE 4 Rumors: Price, Features, Release Date and More (published on 2025-02-13T12:00:00.000Z by macrumors.com)
Summary: Apple is developing a new iPhone SE with a larger 6.06-inch display, all-display design, and Face ID, replacing Touch ID. It's expected to feature a USB-C port, single-lens camera, and A18 chip, supporting Apple Intelligence. The launch is expected in early 2025, with a possible price starting at $429.

[2] iPhone SE 4 & Apple Vision Pro announcements are coming in the next week (published on 2025-02-12T00:00:00.000Z by appleinsider.com)
Summary: Apple is preparing announcements, with the iPhone SE 4 expected by February 21. A smaller announcement is planned for Wednesday, and Apple Vision Pro briefings are set for Friday, possibly related to an operating system or app update. The M4 MacBook Air is expected within weeks, around March. Product briefings for the iPhone SE 4 will occur before its announcement, which is expected soon.

[3] iPhone SE 4 coming soon — 7 biggest rumored upgrades (published on 2025-02-11T12:00:00.000Z by tomsguide.com)
Summary: Apple is expected to launch a new midrange phone, the iPhone SE 4, which will be significantly different from the current model, with rumors suggesting a launch as early as this week or later this spring, featuring a new design, more powerful chipset, Apple Intelligence support, camera upgrade, OLED display, and USB-C port, with a possible price hike to under $500.

[4] I revisited the iPhone SE in 2025 - and the SE 4 may be an even bigger hit (published on 2025-02-11T00:00:00.000Z by zdnet.com)
Summary: Apple is expected to launch a new iPhone SE this week, replacing the current model that launched almost three years ago, with significant upgrades including a 6.1-inch OLED display, Face ID, 48MP main camera, A18 chip, and USB-C port, pricing is expected to be competitive.

[5] iPhone SE 4 Date/Time? (published on 2025-02-10T00:00:00.000Z by forums.macrumors.com)
Summary: Apple users are speculating about the release of the iPhone SE 4, with some expecting it to be announced this week. However, there was no official announcement, and previous SE releases were announced in March. Some think it might be announced on Tuesday or Wednesday, with preorders starting on Friday, while others believe it will be released in March alongside other Apple products. No one knows for sure when it will be released, but the lack of an announcement on Monday has sparked speculation.

[6] iPhone SE 4 rumors: Price, screen, cameras, release date (published on 2025-02-07T12:00:00.000Z by appleinsider.com)
Summary: Apple may announce the fourth-generation iPhone SE as soon as the week of February 14, 2025, with the first units expected to reach customers, and it's expected to feature a design derived from the iPhone 14, with a 6.1-inch OLED display, Face ID, and a 48 MP rear camera.

[7] Long Wait Over For Budget iPhone Buyers As New SE 4 Ready For Release - 9meters (published on 2025-02-07T00:00:00.000Z by 9meters.com)
Summary: The iPhone SE 4 is expected to be announced on February 11, 2025, with pre-sales starting immediately after or on February 14th, and the full release between February 21 and 28th, featuring flagship features at a lower cost, and will be announced through a press release, not a big event.

[8] Here's everything we expect from Apple in February 2025 (published on 2025-02-03T00:00:00.000Z by macworld.com)
Summary: February is expected to be slow for Apple releases, with a possible "direct to the website" launch of the M4 MacBook Air, but no mention of an iPhone SE launch on February 19, and Bloomberg's Mark Gurman suggests the iPhone SE 4 might be held for a bigger event in March.

[9] The MacRumors Show: Latest iPhone SE 4 Rumors (published on 2025-01-31T12:00:00.000Z by macrumors.com)
Summary: The iPhone SE 4 is expected to feature an iPhone 14-style design, 6.1-inch OLED display, Face ID, and USB-C, with a launch expected before iOS 18.4, around March or April, at a price of about $470.

[10] iPhone SE 4 release date: When to expect Apple’s powerful upgrade - 9to5Mac (published on 2025-01-29T00:00:00.000Z by 9to5mac.com)
Summary: Apple is expected to launch the iPhone SE 4 in early 2025, with March being the likeliest date, based on reporting from the supply chain regarding the production timeline and past iPhone SE launch dates (March 18, 2022, April 24, 2020, and March 24, 2017).
----
</News>

Think about your answer and output your final prediction (a number between 0 and 1) with an asterisk at the beginning and end of the decimal (Ex: *<probability>*).

② Model reasoning (<think>):

Okay, let's tackle this. The question is whether Apple will launch an iPhone SE on February 19. The user provided a bunch of news articles from various sources, all dated around early February 2025. The key here is to analyze the information given and determine the probability based on the available data.

First, I need to check the context. The event is scheduled for February 19, and the market will resolve to "Yes" if Apple announces the new iPhone SE during that event. The resolution criteria mention that if the event doesn't happen by that date, it's a "No." Also, the primary source is the official event feed and credible reporting.

Looking at the news articles, there's a mix of information. Some articles, like the one from macrumors.com (article 1) and tomsguide.com (article 3), mention that the iPhone SE 4 is expected to be launched in early 2025, with possible dates around February 21 or March. However, article 2 from appleinsider.com says that the iPhone SE 4 is expected by February 21, with a smaller announcement planned for Wednesday (which would be February 13, 2025, since the current date is February 14). Wait, but the user's current date is February 14, so if the event is on the 19th, that's a week away.

Article 6 from appleinsider.com (published on February 7) mentions that Apple might announce the SE 4 as early as the week of February 14, with first units expected to reach customers. However, article 8 from macworld.com (February 3) says that February is expected to be slow for Apple releases, and there's no mention of an iPhone SE launch on the 19th. Also, Bloomberg's Mark Gurman suggests the SE 4 might be held for a bigger event in March.

Another article, article 5 from forums.macrumors.com (February 10), mentions that some users expect the release this week, but there's no official announcement yet. The article also notes that previous SE releases were in March, so maybe this one is also scheduled for March. However, article 7 from 9meters.com (February 7) says the SE 4 is expected to be announced on February 11, with pre-sales starting on the 14th, and full release between 21-28 Feb. But that's conflicting with the event on the 19th.

Now, the key point is that the event is scheduled for February 19, and the question is whether Apple will announce the SE 4 during that event. However, some articles suggest that the announcement might be earlier, like February 11 or 14, or even in March. The article from appleinsider.com (article 2) mentions that a smaller announcement is planned for Wednesday (February 13), which is a day before the current date (February 14). But the event is on the 19th, so if the announcement is on the 13th, that would be before the event date. However, the user's question is about the event on the 19th, not the announcement date.

But the problem is that the articles are conflicting. Some suggest the SE 4 might be announced in early February, while others think it's in March. The article from 9meters.com (article 7) says the SE 4 is expected to be announced on February 11, with pre-sales starting on the 14th. However, the event is on the 19th, so if the announcement is on the 11th, then the event on the 19th might not be the one for the SE 4. But the user's question is about the event on the 19th, which is a scheduled launch event. However, the articles don't all agree on the exact date.

Also, the article from macrumors.com (article 1) says the launch is expected in early 2025, with a possible price starting at $429. The article from 9to5mac (article 10) says March is the likeliest date, based on past SE launch dates. However, the article from appleinsider.com (article 2) says the SE 4 is expected by February 21, which is after the event date of the 19th. So if the event is on the 19th, and the SE 4 is expected by the 21st, maybe the announcement is on the 19th, but that's not clear.

But the user's question is about the event on the 19th. The problem is that the articles are conflicting. Some suggest the SE 4 might be announced in early February, others in March. The event on the 19th is a scheduled event, but the articles don't all confirm that the SE 4 will be announced there. However, the user's question is whether Apple will launch the SE 4 during that event. The key is that the event is scheduled, but the actual announcement might be on a different date.

But the resolution criteria say that the market will resolve to "No" if the event doesn't occur by the 19th. However, the event is scheduled, so if Apple doesn't announce the SE 4 during that event, it's a "No." But if the SE 4 is announced on the 19th, then it's a "Yes."

But the articles are conflicting. Some suggest the SE 4 might be announced in early February, others in March. The article from appleinsider.com (article 2) says that a smaller announcement is planned for Wednesday (February 13), which is a day before the current date (February 14). However, the event is on the 19th. If the announcement is on the 13th, then the event on the 19th might not be the one for the SE 4. However, the user's question is about the event on the 19th, which is a separate event.

But the problem is that the user's question is about the event on the 19th. The articles don't all agree on whether the SE 4 will be announced there. Some articles suggest that the SE 4 might be announced in early February, but others think it's in March. The article from 9meters.com (article 7) says the SE 4 is expected to be announced on February 11, with pre-sales starting on the 14th. However, the event is on the 19th, so if the announcement is on the 11th, then the event on the 19th might not be the one for the SE 4.

But the user's question is about the event on the 19th. The key is that the event is scheduled, but the actual announcement might be on a different date. However, the user's question is whether Apple will launch the SE 4 during that event. If the event is on the 19th, and the SE 4 is announced during that event, then it's a "Yes." Otherwise, "No."

But the articles are conflicting. Some suggest the SE 4 might be announced in early February, others in March. The article from 9to5mac (article 10) says March is the likeliest date, based on past SE launch dates. However, the article from appleinsider.com (article 2) says the SE 4 is expected by February 21, which is after the event date. So if the event is on the 19th, and the SE 4 is expected by the 21st, maybe the announcement is on the 19th. But that's not certain.

The problem is that the articles don't all agree. Some suggest the SE 4 might be announced in early February, others in March. The event on the 19th is a scheduled event, but the actual announcement might be on a different date. However, the user's question is about the event on the 19th. The answer depends on whether Apple will announce the SE 4 during that event.

Given the conflicting information, the probability is not high. The articles suggest that the SE 4 might be announced in early February, but the event is on the 19th. However, the article from 9meters.com (article 7) says the SE 4 is expected to be announced on February 11, with pre-sales starting on the 14th. That would be before the event on the 19th. Therefore, the event on the 19th might not be the one for the SE 4. However, the article from appleinsider.com (article 2) says that a smaller announcement is planned for Wednesday (February 13), which is a day before the current date (February 14). That might be a different event, not the main SE 4 announcement.

But the user's question is about the event on the 19th. The answer depends on whether Apple will announce the SE 4 during that event. Given the conflicting information, the probability is low. However, the article from 9meters.com (article 7) says the SE 4 is expected to be announced on February 11, which is before the event on the 19th. Therefore, the event on the 19th might not be the one for the SE 4. However, the article from 9to5mac (article 10

③ Gold resolution: No (0) (reference probs — Polymarket 0.92, o1 0.30)

④ Model completion (answer after reasoning):

[no </think> — reasoning ran to the token limit]

Predicted P(Yes) = 1.000; Brier (this question) = 1.000 — exactly what the log/Brier reward penalizes

Track 2 — continuous targets and branching uncertainty

For continuous outcomes the model emits an XML mixture of Betas (K ≤ 4 components, normalized weights), inducing $ p(y) = \sum_k w_k\, \mathrm{Beta}(y;\alpha_k,\beta_k) $ on the range-normalized target, rewarded with the mixture log-likelihood. Unlike the binary case, here the full density is scored — mixtures can genuinely help.

"Branching uncertainty": a synthetic dataset (1,200 train / 300 test) where a discrete latent variable selects between two outcome regimes. A single unimodal Beta cannot express the marginal; a mixture can. Ground-truth branch probabilities and per-branch parameters are stored, so an exact oracle ceiling is computable.

Synthetic branching benchmark — the architectural test

Oracle ceilings: true mixture +0.377 nats mean LL; best possible single Beta (moment-matched) +0.106 nats. The mixture's theoretical advantage is therefore +0.271 nats.

Model	steps	LL ↑	MSE ↓	cov 80%	PIT KS p	% bimodal
Single Beta	200	−0.290	0.095	0.79	0.000	0
Single Beta	800	+0.001	0.085	0.85	0.055	0
Mixture	400	+0.130	0.077	0.86	0.125	100
Mixture	800	+0.223	0.077	0.80	0.82	100

Mixtures capture 82% of the oracle advantage. +0.223 of the available +0.271 nats, with coverage exactly nominal (80% → 0.80) and PIT indistinguishable from uniform (KS p = 0.82). Single Beta at the same 800 steps only reaches +0.001 — the gap is architectural, not a training-budget artifact.

Set-matched check: unlike Track 1, the Track 2 headline rows do not need a matched-subset correction — at s800 both the mixture and single-Beta parse 300/300 test cases (parse rate 1.00), so every continuous metric is already computed over the same complete set. Parse rates only dip at s200 (single-Beta 0.89, mixture 0.98); those are not the reported architectural-comparison rows.

Metaculus continuous questions (real data)

Checkpoint	LL ↑	MSE ↓	cov 80%	mean K	% K≥2
Single Beta s200	−0.988	0.099	—	1	0
Mixture s200	−0.719	0.073	0.66	1.81	80
Mixture s400 (best calibration)	−0.583	0.064	0.76	1.95	95
Mixture s800	−0.461	0.069	0.74	1.76	76

Mixture beats single Beta by −26% MSE and +0.27 nats at matched steps; s800 keeps improving LL but MSE worsens slightly — log-likelihood mildly incentivizes oversharpening. A per-component sharpness bonus (λ=0.1) was ablated and does not help (−0.12 nats on branching, flat on Metaculus).

Open items → and the bridge to project 5

Can RL on the discrete-quantized format match trained Beta? (Would isolate format-vs-training gains.)
Does the mixture advantage persist on genuinely branching real tasks? Mean K is suspiciously similar on branched (1.8–2.0) and unbranched (1.8–1.95) data — general regularizer or task-responsive?
Close the 0.022 Brier gap to ReMax with a 110k-scale training set.

Why project 5 exists. Track 2 proved distributional outputs are valuable but the model must guess mixture weights and parameters directly from text. The natural next step is to let it write a small probabilistic program — "price depends on a coin-flip over dishwasher" — and have an inference engine derive the exact mixture. That is project 5 (Regression-PPL), with Alex's pluck as the engine.

Status: both tracks complete. Last activity: prompt-ablation study, 2026-05-28.

Project 5 — Regression-PPL: mixtures of Gaussians and pluck programs

What it is. The newest line (built 2026-05-28 → today): general distributional regression. The model answers numeric questions — house prices, Fermi estimates, patient survival times — with a full density over the target, scored by the log density $\log p_{\text{model}}(y^{*})$. Two expressiveness levels are compared: an explicit mixture-of-Gaussians XML format, and pluck programs — small probabilistic programs whose exact marginal the pluck engine (Alex's Rust PPL, Python bindings) computes, inducing the density for scoring. Model: Qwen3-4B.

Phase structure. Phases 1–3 complete (Phase 3 trained and evaluated 2026-06-11); Phase 4 baseline scored on-node the same day.

Datasets (6 built, 4 headline)

Dataset	Target	Why it's here	Status
ames_housing (2,930 sales)	log price	discrete features (neighborhood, pool…) gate a continuous price — natural branching	headline + Phase 3
fermieval (3,915 questions)	base-10 exponent	Fermi estimates = products of independent factors	headline + Phase 3
survival (SUPPORT/METABRIC/GBSG)	log time-to-event, right-censored	mixture-cure structure (cured → no event; else T ~ F); censored rows scored as $\log S(y)$	headline + Phase 3
openestimate	per-variable continuous, Gaussian GT	multi-domain numerical estimation	headline (eval only)
metaculus (AutoCast numeric)	normalized position in range	real forecasting questions	leak: resolved pre-cutoff — scalar median log-score +0.244 is implausibly good; within-dataset comparisons only
what_are_the_odds	probabilistic-reasoning QA	distributional reasoning	stubbed: labels need regeneration

Methods — five formats, one comparable score

Every method induces a real density, so log scores are directly comparable across formats:

Method	Output	Induced density
scalar (baseline)	`<answer>V</answer>`	$ \mathcal N(V, \hat\sigma) $, $\hat\sigma$ fit on train residuals — richer formats must beat this
beta	`<beta><alpha>A</alpha><beta>B</beta></beta>`	Beta(A,B) on range-normalized target (Jacobian-corrected)
calibrated_ci	90% interval `[L,U]`	$ \mathcal N\!\bigl(\tfrac{L+U}{2}, \tfrac{U-L}{2\cdot1.645}\bigr) $
mog_xml ← Phase 3 target	`<mixture><component><mu><sigma><w>…`	$ \sum_k w_k \,\mathcal N(y;\mu_k,\sigma_k) $, K ≤ 4
pluck ← Phase 4	Scheme-like program (below)	exact-inference marginal: Gaussian + point-mass components

A pluck program for a mixture-cure survival model — the harness wraps the body in a (query …) scaffold, runs exact inference, and scores the resulting mixture:

(let ((cured (flip 0.3))
      (baseline (gaussian 4.0 1.5)))   ; std-dev, not variance
  (if cured 100.0 baseline))
; → exact marginal: 30% point mass + 70% · N(4.0, 1.5)

Pluck gotchas for the LM. Integer literals are Peano naturals — programs must write 5.0, never 5. No multiplying two Gaussians ((*. c g) with constant c only). No division by a Gaussian. No scientific notation (1e9 panics the parser). No expt. These restrictions are visible in the baseline parse failures below — the base model reaches for exactly the forbidden constructs.

Phase 2 — base-model baselines (n=200/cell, temp 0.7, 2026-05-28)

Log-score mean (primary; higher is better)

dataset	scalar	beta	calibrated_ci	mog_xml
ames_housing	−1.136	−1.343	−4.040	−0.707
fermieval	−11.976	—	−62.710	−6.333
openestimate	−9.812	−8.595	−6.4M	−463k
metaculus (leaky)	−1,862	−11.83	−203k	−35k
survival	−67.022	—	−449.7	−15.488

Log-score median (outlier-robust)

dataset	scalar	beta	cal_ci	mog
ames	−0.687	−0.808	−0.585	−0.493
fermi	−2.073	—	−1.081	−1.172
openest.	−7.119	−8.028	−64.0	−12.26
metaculus	+0.244	−0.287	−21.05	−16.92
survival	−1.899	—	−33.22	−6.259

Coverage of nominal 90% interval

dataset	scalar	beta	cal_ci	mog
ames	0.698	0.870	0.532	0.863
fermi	0.892	—	0.675	0.780
openest.	0.775	0.787	0.085	0.285
metaculus	0.719	0.495	0.167	0.149
survival	0.584	—	0.092	0.343

Readings. MoG-XML wins the mean log score on 3/5 datasets. The mean-vs-median gap on calibrated_ci and mog_xml is itself informative: means are dominated by a small fraction of catastrophically overconfident outputs — the proper scoring rule punishing overconfidence. Calibrated-CI replicates FermiEval's "perception tunnel": asked for a 90% interval, the base LM emits one ~10× too narrow (9% coverage on openestimate/survival). And beta parses only ≈50% of the time (the model writes "I'd say Beta(3,5)" as prose) — so beta is a format-discipline test, not a scoring-rule test; it is excluded from Phase 3 training.

Example rollouts — base model (click to expand)

ames_housing · base MoG-XML — parse rate 0.98, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating the sale price of a residential property in Ames, Iowa (sales 2006-2010).

Property features:
- Neighborhood: Somerset
- MS_Zoning: Residential_Low_Density
- House_Style: One_Story
- Year_Built: 2008
- Year_Remod_Add: 2008
- Overall_Qual: Excellent
- Overall_Cond: Average
- Gr_Liv_Area: 1506
- Total_Bsmt_SF: 1498
- Garage_Type: Attchd
- Garage_Cars: 2
- Full_Bath: 2
- Half_Bath: 0
- Bedroom_AbvGr: 2
- Pool_QC: No_Pool
- Fireplaces: 1
- Lot_Area: 14587
- Central_Air: Y

Provide your estimate of the natural log of the sale price in USD (typical values 10.5 to 13.5).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the sale price for this property in Ames, Iowa from 2006-2010. The user wants a mixture of up to four Gaussian components. Let me start by recalling the features provided.

First, the neighborhood is Somerset. I remember that Somerset is a higher-end area, so that might contribute to a higher price. The MS_Zoning is Residential_Low_Density, which is good for a home. The house is one-story, built and remodeled in 2008, so it's a newer home. The overall quality is excellent, but the condition is average. That's a bit conflicting. A high quality but average condition might mean the home is well-maintained but not in top shape.

The living area is 1506 sq ft, and the basement is 1498. That's a large size. The garage is attached with 2 cars, which is good. Two full baths, no half baths, two bedrooms above ground. No pool, but there's a fireplace. The lot is 14,587 sq ft, which is pretty big. Central air is present, which is a plus.

Now, considering the sale price log. Typical values are 10.5 to 13.5. Let's think about the factors. The neighborhood (Somerset) might increase the log price. The size of the house (both living and basement) is significant. The year built is 2008, which is recent, so maybe that's a factor. The overall quality is excellent, which might add to the price, but the condition is average. That's a bit confusing. Maybe the excellent quality is due to the home being well-maintained, but maybe not the best in terms of condition. The fireplace and central air are also positives.

I need to model this with a mixture of Gaussians. Let's consider different scenarios. One scenario could be a high-quality home in a good neighborhood with good size and features, leading to a higher log price. Another scenario might be a home with good size but maybe lower quality or neighborhood, leading to a lower log price. Maybe the excellent quality and good neighborhood could be one component, and the average condition might be another. Also, the size and features like central air and fireplace could be factors.

So, possible components: a high component for a well-located, large home with good features, maybe around 12.5 with a small sigma. Another component for a slightly lower price due to average condition, maybe around 11.5. But since the condition is average, but the quality is excellent, maybe that's a mix. Alternatively, maybe two components: one for the high-quality, good neighborhood, and another for the average condition but good size. Wait, the user said up to four components. Maybe another component for the size and basement area, and another for the presence of central air and fireplace. But I need to think about which factors are most influential.

Alternatively, the main factors are neighborhood, size, and quality. Since the neighborhood is Somerset, which is high, that's a big factor. The size is large, which is a big factor. The quality is excellent, but condition is average. Maybe the excellent quality adds a bit, but the average condition might not. So, the main components could be a high-value component (maybe 12.5 with low sigma) and a lower component (maybe 11.5 with higher sigma). But I need to check if there are other factors. The presence of central air and fireplace, which are positive, but maybe they are already included in the size and quality. Alternatively, maybe the fireplace adds a little to the log price.

Alternatively, considering that the sale price is in the range of 10.5 to 13.5, and the property is in a good neighborhood with good size, maybe the log price is around 12.5. But with uncertainty, so maybe two components: one around 12.5 with a small sigma, and another around 11.5 with a larger sigma. But the user allows up to four components. Maybe three components: one for the high-quality neighborhood, one for the size, and one for the average condition. Wait, but I need to think of different scenarios. For example, if the home is in a good neighborhood, has good size, and excellent quality, that's one scenario. If the home is in a good neighborhood but has average condition, that's another. Maybe the presence of central air and fireplace could be another component. But I need to make sure that the mixture captures the uncertainty.

Alternatively, the main factors are the neighborhood (Somerset) which is a high value, the size (large living area and basement), and the quality (excellent). So maybe the main component is around 12.5, but there could be another component if the condition is average, which might lower the log price slightly. So two components: one at 12.5 with a small sigma, and another at 11.8 with a larger sigma. The weights could be 0.6 and 0.4. But I need to check if that's reasonable. Alternatively, maybe three components: one for the high-quality, one for the average condition, and one for the size. But I'm not sure. The user said up to four, but maybe two is sufficient here. Let me go with two components. The first component is 12.5 with a small sigma, and the second is 11.8 with a larger sigma. The weights would be 0.7 and 0.3. That way, the main component is the high value, but there's a lower component for uncertainty. Alternatively, maybe the average condition is a bit lower, but the size and quality are high. Maybe the correct answer is a mixture of 12.3 and 11.8. But I need to think about the typical log prices. If the typical is 10.5-13.5, and this property is in a good neighborhood, large size, excellent quality, maybe the log price is around 12.5. But with some uncertainty. So the mixture would have two components: one around 12.5 with low sigma and another around 11.5 with higher sigma. But I need to make sure the numbers are reasonable. Let me check the possible values again. The natural log of the sale price. For example, if the sale price is $200,000, the log is about 12.2. If it's $300,000, it's around 12.7. So 12.5 would be in the middle. The average condition might lower it by 0.5 to 0.7, so around 11.8. So the mixture would be two components: 12.5 (weight 0.7) and 11.8 (weight 0.3). The sigma for the first is small, like 0.2, and the second is larger, like 0.5. That seems plausible.

③ Gold label: y = 12.4842 (log USD)

④ Model completion (answer after reasoning):

<mixture>
  <component><mu>12.5</mu><sigma>0.2</sigma><w>0.7</w></component>
  <component><mu>11.8</mu><sigma>0.5</sigma><w>0.3</w></component>
</mixture>

Parsed → induced: 0.70·N(12.50, 0.20) + 0.30·N(11.80, 0.50); log-score 0.396

✗ Overrun — reasoning overran, no answer

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating the sale price of a residential property in Ames, Iowa (sales 2006-2010).

Property features:
- Neighborhood: College_Creek
- MS_Zoning: Residential_Low_Density
- House_Style: Two_Story
- Year_Built: 1997
- Year_Remod_Add: 1997
- Overall_Qual: Above_Average
- Overall_Cond: Average
- Gr_Liv_Area: 1771
- Total_Bsmt_SF: 864
- Garage_Type: Attchd
- Garage_Cars: 2
- Full_Bath: 2
- Half_Bath: 1
- Bedroom_AbvGr: 3
- Pool_QC: No_Pool
- Fireplaces: 0
- Lot_Area: 13142
- Central_Air: Y

Provide your estimate of the natural log of the sale price in USD (typical values 10.5 to 13.5).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

First, the neighborhood is College_Creek. From what I know, College_Creek is a higher-end neighborhood, so that might contribute positively to the price. Then the MS_Zoning is Residential_Low_Density, which is good for higher prices. The house is a Two_Story, built in 1997, same as the remod. Overall Qual is Above Average, which is a positive factor. Gr_Liv_Area is 1771 sq ft, which is decent. Total Bsmt SF is 864, so basement is a part of the house. Garage is attached, 2 cars, which is standard. Full Baths 2, Half 1. Bedrooms above ground 3. No pool, no fireplaces. Lot area 13,142 sq ft, which is decent. Central Air is present.

Now, I need to think about how these features affect the sale price. The natural log of the price is typically modeled in regression analysis, so I need to estimate the log(price) based on these features. The typical range given is 10.5 to 13.5, so the estimate should be within that range.

Let me consider the components. The key factors here are the neighborhood (College_Creek), which is likely a high-value area. The overall quality is above average, so that's a plus. The size of the house is 1771 sq ft, which is a significant factor. The central air is a plus. The garage is attached, which is a positive feature. The number of bedrooms and bathrooms might also contribute. However, the presence of a pool is a negative, but here it's no pool. The lot size is 13,142 sq ft, which is a good size, but not extremely large.

But I need to model this as a mixture of Gaussians. Maybe the main component is the base value based on the house size and location. Then, other components could be based on different factors like neighborhood, overall quality, or maybe the presence of central air. However, since the user said up to four components, I should consider if there are different scenarios that could affect the price.

For example, the neighborhood (College_Creek) might be a strong component. The overall quality is above average, which is a positive. The house size is 1771, which is a good value. The central air is a positive. So maybe the main component is around 12.0 with a small sigma, but considering that the house is in a good neighborhood and has good features, maybe the mean is higher. Wait, but the typical range is 10.5 to 13.5, so 12.0 is in the middle. But I need to think about possible variations.

Alternatively, maybe there's a component for the neighborhood's effect, another for the house size, another for the overall quality, and another for central air. But since the user wants a mixture, I need to model the uncertainty. For example, if the house is in a good neighborhood, maybe the log price is higher. But if the overall condition is average, that might bring it down. However, the Overall Qual is above average, so that's a positive.

Wait, the Overall Qual is Above Average, which is a positive factor. The Overall Cond is Average, which is neutral. So maybe the main component is based on the house size and neighborhood. So the first component could be around 12.0 with a small sigma, say 0.3. Then another component for the neighborhood effect, maybe 12.5 with a slightly larger sigma. But I need to check if there are other factors. The central air is present, which is a positive, so maybe another component at 12.2 with a sigma of 0.4. But I have to make sure the components don't overlap too much. Alternatively, maybe three components: one for the base, one for the neighborhood, one for the size, and another for the quality. But since the user allows up to four, I can have four components. However, the answer needs to be concise.

Alternatively, considering that the house is in College_Creek, which is a high-value area, maybe the first component is around 12.5 with a small sigma. Then, the house size and overall quality might bring it down a bit. But I think the main component is the neighborhood. However, the overall quality is above average, which is a positive. Maybe the main component is 12.2 with a sigma of 0.3. Then another component for the neighborhood's effect, say 12.8 with sigma 0.4. But I need to make sure that the weights sum to 1 when normalized. But the user said weights don't need to sum to 1.

Alternatively, perhaps the main component is 12.0 with a sigma of 0.3, and another component for the neighborhood effect, 12.2 with sigma 0.4, and two more for other factors. But I need to think of possible scenarios. For example, if the house is in a good neighborhood, it's higher. If the overall quality is average, maybe it's a bit lower. But the Overall Qual is above average, so that's a positive.

Alternatively, maybe the main component is around 12.5, with a small sigma, and another component for the house size, which is 1771. Let me think about the typical log price. For a house in College_Creek, with a size of 1771, maybe the log price is around 12.5. But given that the overall condition is average, maybe it's a bit lower. Maybe 12.3 with a sigma of 0.3. Another component could be for the presence of central air, which is a positive, maybe 12.4 with sigma 0.4. Then another component for the neighborhood, maybe 12.6 with sigma 0.3.

But I need to ensure that the mixture components are reasonable. The user might expect that the main component is around 12.2-12.5. Let me check some references. For example, in the Ames dataset, the log price for similar features. But since I can't access that, I have to rely on general knowledge.

Assuming that the house is in a good neighborhood, with a decent size, central air, and above average quality, the log price is likely around 12.2. But there's uncertainty. Maybe the first component is 12.2 with sigma 0.3, another component for the neighborhood effect, 12.5 with sigma 0.4, and two more components for other factors. But perhaps it's better to have three components. Alternatively, maybe two components: one for the main effect (around 12.2 with small sigma) and another for the neighborhood, which is 12.5 with a larger sigma. But the user allows up to four components.

Alternatively, maybe the main component is 12.3 with sigma 0.3, and another component for the neighborhood, 12.5 with sigma 0.4, and another for the house size, 12.2 with sigma 0.3, and another for the central air, 12.4 with sigma 0.4. But I need to make sure the components are distinct enough.

Alternatively, considering that the house is in College_Creek, which is a high-end area, the main component could be 12.5 with a small sigma. Then, considering that the overall condition is average, maybe another component at 12.2 with a larger sigma. But this might not be the best approach.

I think the best approach is to have a main component around 12.2 with a small sigma, and another component for the neighborhood effect, 12.5 with a slightly larger sigma. Then, maybe two more components for other factors like house size and central air. But I need to decide. Since the user allows up to four, maybe three components would be sufficient.

Alternatively, the main component is 12.2 with sigma 0.3, and another component for the neighborhood, 12.5 with sigma 0.4, and two more components for the house size and central air. But I think the main components are the neighborhood and the house size.

Alternatively, given that the house is in College_Creek, which is a high-value neighborhood, and the house has good size and features, the log price is likely around 12.3. But with some uncertainty. So, maybe the mixture is three components: 12.2, 12.4, 12.6 with varying weights. But I need to make sure that the components are plausible.

In the end, I think the best approach is to have three components: one around 12.2 with a small sigma, another around 12.5 with a slightly larger sigma, and a third around 12.3 with a moderate sigma. But I need to check the numbers again. Given the typical range of 10.5 to 13.5, and the features, I think the estimate is around 12.2-12.5. So, I'll create three components: 12.2, 12.3, and

③ Gold label: y = 12.2816 (log USD)

④ Model completion (answer after reasoning):

[no </think> — reasoning overran the token budget; no answer emitted]

Outcome: no answer block — <think> consumed the token budget → parse failure (reward floor).

survival · base MoG-XML — parse rate 0.71, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating a time-to-event outcome in the gbsg survival cohort. The feature columns are anonymized; reason from the marginal distribution and any plausible scale you can infer from the cohort name.

Patient features:
  x0: 1
  x1: 1
  x2: 0
  x3: 51
  x4: 1
  x5: 167
  x6: 109

Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, let's tackle this problem. The user is asking for an estimate of the natural log of the time to event in the GBSG cohort. The features given are x0 to x6, but they're anonymized. The cohort name is GBSG, which I remember is the German Breast Cancer Study Group. From what I recall, GBSG data is often related to breast cancer survival, and time-to-event outcomes are usually measured in months or years.

First, I need to figure out the plausible scale. Since GBSG is a breast cancer study, the time to event is likely in months. But the user says to use the same time unit as the cohort. Wait, the question says to express the answer in the same time unit as the cohort, which for GBSG is months. Wait, but the problem says "days for SUPPORT, months for METABRIC and GBSG." So GBSG uses months. Therefore, the log of time would be in log(months). But the user wants the natural log of the time to event in the same time unit as the cohort. So if the time is in months, then the log is ln(months).

But how do I estimate the log of the time to event? The features given are x0 to x6. However, the problem says the feature columns are anonymized. So I can't know their actual meaning. But the user says to reason from the marginal distribution and any plausible scale. So maybe I need to infer the possible values based on the GBSG study.

Looking at the features: x0 is 1, x1 is 1, x2 is 0, x3 is 51, x4 is 1, x5 is 167, x6 is 109. These numbers might be variables that could relate to patient characteristics. For example, x3 is 51, x5 is 167, x6 is 109. If these are measurements like age, maybe? But age is usually in years. But 51 could be age in years, 167 might be something else. Alternatively, maybe these are survival times or other metrics. But since the features are anonymized, I can't be sure.

Alternatively, maybe the features are related to the outcome. For example, in survival analysis, features like tumor size, node count, etc., could be variables. But without knowing what each x is, it's hard to say. But the question mentions that the answer should be a mixture of up to 4 Gaussian components. So maybe the log of the time to event has a distribution that can be approximated by a mixture of Gaussians.

In GBSG studies, survival times for breast cancer can vary. For example, in some cases, the median survival might be around 5-10 years, but that's in months. If the time is in months, then the median might be around 60-120 months. Taking the natural log of that would be ln(60) ≈ 4.09, ln(120) ≈ 4.78. But if the survival times are longer, maybe up to 200 months (about 16.6 years), ln(200) ≈ 5.3. But I'm not sure. However, the features given are x3=51, x5=167, x6=109. These numbers could be related to the outcome.

Alternatively, maybe the features are not the survival times but other variables. For example, x3=51 could be a count of something. But the problem states that the user is to estimate the log of the time to event. So perhaps the answer is based on the marginal distribution of the time to event in GBSG.

In the GBSG study, the survival times for breast cancer patients could vary. Let's say the average survival time is around 10-15 years, which is 120-180 months. The log of that would be around 4.78 to 5.19. But if there are different subgroups, maybe the distribution has multiple modes. For example, some patients might have shorter survival, others longer. So a mixture of Gaussians would make sense.

Assuming that the log of the time to event has a mixture of two components: one centered around 4.5 (for shorter survival) and another around 5.5 (for longer survival). But the problem allows up to 4 components. Maybe three components? Let me think. If the data has a bimodal distribution, maybe two components. But the user says to use up to four. However, the features given are x3=51, x5=167, x6=109. These numbers might be related to the time to event. Wait, x3 is 51, which could be months. If that's the case, then ln(51) is about 3.93. But that's one data point. However, the features are not the survival times but other variables. So maybe the actual survival times are not directly given by these features.

Alternatively, maybe the features are the survival times. But the problem says that the features are anonymized, so the numbers are not the actual survival times. So I have to infer based on the GBSG cohort's typical time-to-event outcomes.

In GBSG, the survival time is often measured in months. For example, in the GBSG trial, the median survival time for patients with high-risk breast cancer might be around 5-6 years (60 months). Taking the natural log, that would be ln(60) ≈ 4.09. However, if the data has a longer tail, maybe some patients survive longer, so the log might have a higher mean. Suppose the distribution is a mixture of two Gaussians: one centered at 4.5 (for shorter survival) and another at 5.5 (for longer survival). The weights could be 0.6 and 0.4, respectively. Alternatively, three components: maybe one at 4.0, one at 4.8, and one at 5.5, with weights 0.3, 0.5, 0.2. But without more data, it's hard to say. Alternatively, maybe the log of the time is normally distributed with a mean around 4.5 and standard deviation 0.5. But the problem allows for a mixture.

Alternatively, considering that the features given are x3=51, x5=167, x6=109. If these are survival times in months, then ln(51) ≈ 3.93, ln(167) ≈ 5.12, ln(109) ≈ 4.70. These could be samples from the distribution. But since the features are not the survival times, maybe these numbers are related to other variables. However, the user is asking for the log of the time to event, so maybe the answer is based on the GBSG's typical survival times.

In conclusion, the natural log of the time to event in months for GBSG is likely to have a distribution that could be modeled as a mixture of two Gaussians. For example, a component around 4.5 with a standard deviation of 0.5 and another around 5.5 with a standard deviation of 0.7, with weights 0.6 and 0.4. But since the problem allows up to four components, maybe three components. However, without more information, two components seem reasonable.

③ Gold label: y = 2.3589 (log time) [right-censored]

④ Model completion (answer after reasoning):

<mixture>
  <component><mu>4.5</mu><sigma>0.5</sigma><w>0.6</w></component>
  <component><mu>5.5</mu><sigma>0.7</sigma><w>0.4</w></component>
</mixture>

Parsed → induced: 0.60·N(4.50, 0.50) + 0.40·N(5.50, 0.70); log-score -0.000

✗ Overrun — reasoning overran, no answer

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating a time-to-event outcome in the metabric survival cohort. The feature columns are anonymized; reason from the marginal distribution and any plausible scale you can infer from the cohort name.

Patient features:
  x0: 5.97
  x1: 8.215
  x2: 10.58
  x3: 5.353
  x4: 1
  x5: 0
  x6: 0
  x7: 1
  x8: 66.04

Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, let's tackle this problem. The user is asking for the natural log of the time to event in the METABRIC cohort, which I know is a breast cancer dataset. The features provided are x0 to x8, but they're anonymized. I need to infer the possible scale based on the cohort name.

First, METABRIC is a breast cancer study, so survival times are typically measured in months. But I should confirm that. The features given are numerical, but without knowing their exact meaning, I have to rely on the cohort's standard.

Looking at the numbers: x0 is 5.97, x1 is 8.215, x2 is 10.58, x3 is 5.353, x4 is 1, x5 is 0, x6 is 0, x7 is 1, x8 is 66.04. The last feature, x8, is 66.04. If this is a survival time, maybe in months, then 66 months is about 5.5 years. But METABRIC usually has survival times in months. However, the question says to estimate the natural log of the time to event.

Wait, but the features are not the survival times themselves. The user is asking for the natural log of the time to event, which would be the log of the survival time. But how do I estimate that? The problem says to reason from the marginal distribution and plausible scale.

Since the cohort is METABRIC, survival times are usually in months. But the features given are not the survival times. The features are other patient features. So maybe the features are related to the survival time. But the user is asking for the log of the time to event, which is a single value. Wait, maybe the features are the actual survival times? But the numbers like 66.04 could be months. However, the first four features are around 5-10, which might be in months as well. But that seems a bit low. Alternatively, maybe the features are other variables, and the time to event is something else.

Alternatively, maybe the features are the survival times. If x8 is 66.04, and that's in months, then the log of that is ln(66.04) ≈ 4.19. But the other features are lower. However, the question says to estimate the log of the time to event, so maybe the features are the survival times. But the problem says the features are anonymized, so I can't know for sure.

Alternatively, maybe the features are related to the survival time. For example, in METABRIC, some features might be related to clinical variables that influence survival. But without knowing the specific features, I have to make an assumption.

Alternatively, the problem might be that the features are the survival times, and the user is asking for the log of the survival time. For example, if x8 is 66.04 months, then ln(66.04) is about 4.19. But the other features are x0 to x7. If those are also survival times, then their logs would vary. However, the user is asking for the estimate of the natural log of the time to event. Maybe the features are the actual survival times, and the user is looking for the log of the survival time. But since the features are anonymized, perhaps the values are in months.

Alternatively, maybe the features are not the survival times, but the problem is to estimate the log of the survival time based on the features. But without knowing the model or the relationship between the features and the survival time, this is tricky.

Alternatively, maybe the features are the survival times, and the user is asking for the log of the survival time. For example, if x8 is 66.04, then ln(66.04) is approximately 4.19. Then the other features would be other survival times. But the problem says to provide a mixture of Gaussian components. So perhaps the log of the survival times in METABRIC are normally distributed.

METABRIC has survival times in months. The average survival time for METABRIC is around 30-40 months. The log of that would be around 3.4 to 3.7. But the data given includes x8 as 66 months, which is ln(66) ≈ 4.19. So maybe the distribution of log survival times is centered around 3.5-4.0, with some variation.

But the features given are x0 to x8. If x8 is 66 months, then ln(66) ≈ 4.19. But maybe the other features are in months as well. For example, x0 is 5.97 months, ln(5.97) is 1.79. But that seems too low. Alternatively, if the features are in days, then x8 is 66.04 days, which is about 2 months, but that seems inconsistent with the other features.

Alternatively, maybe the features are in months, but the survival time is in months. So the log of the survival time would be the log of the months. So if the survival time is around 66 months, the log is about 4.19. However, the other features are lower.

But the question is to estimate the natural log of the time to event. So maybe the features are the survival times, and the answer is to take the log of each feature. However, the user is asking for the estimate of the log, which would be a mixture of Gaussian components. So perhaps the log of the survival times in METABRIC are normally distributed.

Looking up METABRIC data, the survival times are usually in months. The average survival time is around 30-40 months. The log of that is around 3.4-3.7. However, there are some patients with longer survival, maybe up to 100 months. So the log could be up to ln(100) ≈ 4.6.

But the features given include x8 as 66.04, which is 66 months, ln(66) ≈ 4.19. The other features are lower. So maybe the log of the survival time is around 3.5 to 4.5. But the problem is to estimate the log of the time to event.

Alternatively, maybe the features are not the survival times. But the problem states that the feature columns are anonymized. So perhaps the features are other variables, but the time to event is something else. However, the user is asking for the natural log of the time to event.

Wait, maybe the features are the survival times. The problem says "the feature columns are anonymized" but the question is to estimate the natural log of the time to event. So perhaps the features are the survival times, and the user is to compute the log of each feature.

But the question is to provide a mixture of up to 4 Gaussian components. So maybe the log of the survival times in METABRIC are normally distributed. For example, if the survival times are in months, then the log would have a mean around 3.5 and a standard deviation of 0.5. But that's a guess.

Alternatively, since x8 is 66.04, which is ln(66.04) ≈ 4.19, and other features are lower, maybe the distribution has a mean around 4.0 with a standard deviation of 0.5. But that's just a guess.

Alternatively, the problem might be that the features are not the survival times, but the log of the survival times. Then, the features are already in log scale. But the question is to estimate the log of the time to event.

This is confusing. But given that the cohort is METABRIC, and survival times are in months, the log of the survival time would be in ln(months). So if the survival times are around 30-100 months, the log would be around 3.4 to 4.6.

But the given features include x8 as 66.04, which is ln(66.04) ≈ 4.19. So maybe the distribution of the log survival times is centered around 4.0 with a standard deviation of 0.5. So a Gaussian component with mu=4.0, sigma=0.5, weight=0.8. But maybe there's a second component for shorter survival times.

Alternatively, perhaps the features are in days. For example, x8 is 66.04 days, which is about 2.2 months. Then the log would be ln(66.04) ≈ 4.19, but that seems inconsistent with the other features.

Alternatively, maybe the features are in months, but the time to event is in months. So the log of the time to event is the log of the months. So if the features are survival times, then the log would be as calculated.

Given that, and considering METABRIC's survival times, I think the log of the time to event is around 3.5 to 4.5. Maybe a mixture of two Gaussians: one centered at 3.5 with sigma 0.5, and another at 4.0 with sigma 0.5. But I'm not sure. Alternatively, maybe only

③ Gold label: y = 4.9178 (log time)

④ Model completion (answer after reasoning):

[no </think> — reasoning overran the token budget; no answer emitted]

Outcome: no answer block — <think> consumed the token budget → parse failure (reward floor).

fermieval · base MoG-XML — parse rate 0.66, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: How deep in cm is the deepest part of the ocean?

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, so I need to estimate the order of magnitude of the deepest part of the ocean in centimeters. Let me start by recalling that the Mariana Trench is the deepest part of the ocean. The Challenger Deep is the deepest known point, right? I think the depth is around 11,000 meters. Let me check that. Wait, yes, about 10,994 meters. So converting that to centimeters, since 1 meter is 100 cm, that would be 10,994 * 100 = 1,099,400 cm. So that's approximately 1.1 million centimeters. The order of magnitude here would be 10^6 cm, since 1.1 million is 1.1 x 10^6.

But wait, the question says to provide the base-10 exponent as an integer. So 1.1 million is 10^6. However, the answer might have some uncertainty. Maybe I should consider if there's a different possible depth. For example, if someone thought the depth was 10,000 meters, that would be 10^6 cm as well. But if there's a different scenario, like if the depth was 10,000 meters, that's still 10^6 cm. Alternatively, if someone thought it was 10,000,000 cm, that's 10^7, but that's 100,000 meters, which is way too deep. So the correct answer is definitely around 10^6 cm.

But the problem says to use a mixture of up to 4 Gaussian components. So maybe there's some uncertainty. For example, maybe the depth is sometimes considered as 10^6, but maybe there's a possibility of 10^5 or 10^7? But that seems unlikely. The Mariana Trench is about 11 km, which is 1.1 x 10^6 cm. So the main component would be 6. But maybe there's some uncertainty. For instance, if someone thought the depth was 10,000 meters (10^6 cm) or 10,000,000 cm (10^7 cm), but that's not really a different scenario. Alternatively, if there's a different ocean trench, but I think the Mariana Trench is the deepest. So maybe the answer is 6 with a small sigma. Alternatively, maybe there's a 10^5 cm (10 km) if someone miscalculates, but that's not correct. So maybe the main component is 6 with a small sigma, and maybe a 5 with a smaller weight if there's some uncertainty. But I think the correct answer is 6, so the mixture would be 6 with sigma 0.5 and weight 1. But the problem says to use up to 4 components. Maybe 6 is the main, but if there's some uncertainty, say, 5.5, 6, 6.5, and 7. But that's not right. Wait, the actual depth is 1.1 million cm, which is 1.1 x 10^6, so the exponent is 6. So the order of magnitude is 10^6. Therefore, the answer is 6. So the mixture would be one component with mu=6, sigma=0.5, and weight 1. But maybe the user wants multiple components. However, the problem says to use up to 4 components when the answer is uncertain across different scenarios. But in this case, the answer is pretty certain. The depth is around 10^6 cm. So maybe the mixture is just one component. However, the problem says to use multiple components when uncertain. But since the answer is certain, maybe just one component. But the example given in the problem shows multiple components. Wait, the user says: "Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios." So if there's uncertainty, like if the depth could be in different ranges, then use multiple. But in this case, the depth is known to be around 10^6 cm, so maybe the answer is 6 with a small sigma. So the mixture would be a single component with mu=6 and sigma=0.5. But the problem says to use up to 4. Maybe the user wants me to consider that maybe someone might think it's 10^5 or 10^7, but that's not the case. Alternatively, maybe the Mariana Trench is sometimes cited as 11 km, which is 1.1 x 10^6 cm, so the order of magnitude is 10^6. Therefore, the answer is 6. So the mixture would be one component with mu=6, sigma=0.5, and weight 1. But perhaps the user wants me to consider that maybe the depth is sometimes considered as 10^6 or 10^7, but that's not correct. So maybe the correct answer is 6 with a small sigma. Therefore, the final answer is 6.

③ Gold label: y = 6.0000 (log10)

④ Model completion (answer after reasoning):

<mixture>
  <component><mu>6</mu><sigma>0.5</sigma><w>1</w></component>
</mixture>

Parsed → induced: 1.00·N(6.00, 0.50); log-score -0.226

✗ Rejected — program/mixture emitted but rejected

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: How many tetrominoes (Tetris pieces) would it take to complete an ideal 40-line sprint?

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, let's tackle this Fermi question. The user is asking how many tetrominoes it would take to complete an ideal 40-line sprint in Tetris. First, I need to recall what a 40-line sprint means. In Tetris, a "sprint" is when you clear lines quickly, and a 40-line sprint would mean achieving 40 lines in a row. But the question is about how many tetrominoes are needed to complete that.

Wait, maybe the question is a bit different. Wait, the user says "complete an ideal 40-line sprint." So, maybe it's about how many tetrominoes are used in a 40-line run. But how does that work? Each line cleared requires certain tetrominoes. But the problem is that in Tetris, each line cleared is made up of different tetrominoes. However, the exact number of tetrominoes used depends on the number of lines and how they are arranged.

But maybe the key is to figure out how many tetrominoes are used to clear 40 lines. Each line is 10 blocks, so 40 lines would be 400 blocks. But each tetromino covers 4 blocks. So, if you have 400 blocks, divided by 4, that would be 100 tetrominoes. But wait, that's assuming each tetromino is used exactly once and that there's no overlapping. But in reality, when you clear lines, you might have some pieces that are used in multiple lines, but in a perfect scenario, maybe each tetromino is used once. So maybe 100 is the answer. But the problem is that in Tetris, you can't just have a perfect arrangement where every piece is used once. Wait, but the question says "ideal 40-line sprint," which might imply that the player is able to clear lines in a way that maximizes the number of tetrominoes used, or maybe it's the minimum number required to clear 40 lines. Hmm, maybe the answer is 100.

But wait, maybe the question is about how many tetrominoes are needed to complete the 40 lines, considering that each line requires a certain number of tetrominoes. For example, each line is made up of 4 blocks, but each tetromino is 4 blocks. So, if you have 40 lines, that's 400 blocks. Each tetromino is 4 blocks, so 400 /4 = 100. So, the answer would be 100. So the order of magnitude is 10^2. But maybe there's a different way to think about it. For example, in a 40-line sprint, the number of tetrominoes might be more or less depending on the scoring. But the question says "ideal," so maybe it's the minimal number. Therefore, 100, which is 1e2. So the exponent is 2. But I need to consider possible variations. Maybe the answer is different. For example, if the player is using a lot of tetrominoes, but maybe the answer is 100. Alternatively, maybe there's a different way to calculate it. Let me think again.

In Tetris, each line cleared requires 10 blocks. To clear 40 lines, that's 400 blocks. Since each tetromino is 4 blocks, 400 divided by 4 is 100. So that's 10^2. So the order of magnitude is 10^2. But maybe the answer is different. For example, if the player is using more tetrominoes because of the way they are arranged, but in an ideal scenario, maybe that's not the case. So the answer is 100, so the exponent is 2. But the user wants a mixture of up to 4 Gaussian components. So maybe there's some uncertainty. For example, if the ideal 40-line sprint requires more or less. Let's say there's a 10% chance of 100, 20% of 100, 30% of 100, etc. Wait, but the question is about the order of magnitude. So if the answer is 100, then the exponent is 2. But maybe the answer could be 10^2, 10^3, or something else. Wait, but 100 is 1e2, so the base-10 exponent is 2. So the answer is 2. But I need to consider that maybe the actual number is different. For example, if you have to account for the fact that some tetrominoes are used in multiple lines, but in an ideal scenario, maybe that's not the case. Alternatively, maybe the answer is 100. So the mixture would be a single component with mu=2, sigma=0, weight=1. But the user said to use up to 4 components when uncertain. So perhaps the answer is 100, but maybe there's some uncertainty. For example, if the ideal sprint requires more due to some factors, or less. But given that the calculation is straightforward, maybe it's 100. So the answer is 2. So the mixture would be a single component with mu=2, sigma=0. But the user said to use multiple components when uncertain. But maybe there's a range. For example, maybe the answer could be between 100 and 200, so mu=2, sigma=0.5, or something. But I think the answer is 100, so exponent 2. So the mixture would be a single component with mu=2 and sigma=0. So the answer is 2.

③ Gold label: y = 2.0000 (log10)

④ Model completion (answer after reasoning):

<mixture>
  <component><mu>2</mu><sigma>0</sigma><w>1</w></component>
</mixture>

Outcome: answer emitted but the scorer rejected it → counted as a parse failure (reward floor).

✗ Overrun — reasoning overran, no answer

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: Compared to all humans who have ever existed, what fraction of them are competing in today’s tournament?

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, let's tackle this Fermi question. The question is asking for the fraction of all humans who have ever existed that are currently competing in today's tournament. First, I need to estimate the total number of humans who have ever lived. I remember that the human population has grown exponentially, but for a rough estimate, maybe around 10^11 (100 billion) people. Wait, actually, the total number is estimated to be around 10^11, but I think the exact figure might be a bit higher. Let me think: as of now, the current population is about 8 billion, and if we go back to the start of humanity, say 200,000 years ago, the population would have been much lower. But the total number is often cited as around 10^12, but I'm not sure. Wait, maybe it's safer to say 10^11. Alternatively, some sources say around 10^12, but I need to check.

Wait, the current population is about 8 billion. If we assume that the average population over the centuries was much lower, but the total number is the sum of all people who have ever lived. Let me recall: the total number of humans is roughly 10^11 (100 billion) or 10^12 (1 trillion)? I think it's more like 10^11. Let me think: if the population has grown from a few hundred thousand to 8 billion, but that's only the last few thousand years. The total number would be more like 10^11. For example, if we take the average population over the entire history as maybe 10^5 (a million) and multiply by the time, say 10^6 years (1 million years), that's 10^11. So that gives me 10^11 as a rough estimate.

Now, the number of people competing in today's tournament. Let's assume a tournament like the Olympics, but the question is about "today’s tournament"—maybe a single event, like the Olympics. The number of participants in the Olympics is around 10^3 (thousands), but if it's a larger tournament, maybe 10^4. However, the question says "competing in today’s tournament"—maybe it's a single event, so maybe a few thousand. But if it's a global tournament, maybe 10^4. However, this is just one event. Alternatively, maybe the question is considering all current competitors in all tournaments, which might be more. But the problem is that the question is vague. Alternatively, if we take the number of people currently participating in any tournament, but that's a large number. However, the question says "today’s tournament"—so maybe a single tournament. Let's assume that the number of competitors is around 10^3 (like the Olympics has about 10,000 athletes, but maybe that's a rough estimate). Let's say 10^3.

So the fraction would be 10^3 / 10^11 = 10^-8. But that's a very small fraction. However, maybe the total number of humans is higher. If the total is 10^12, then 10^3 / 10^12 = 10^-9. But I need to check my estimates again.

Wait, the total number of humans is often cited as around 10^12. For example, if we consider that the population has been growing exponentially, and the current population is 8 billion, but over the entire history, the total number is estimated to be around 10^12. For example, some sources say that the total number is about 100 billion, but others say 1 trillion. Let me think: if the population has been around 100 million for a long time, then over 1 million years, that's 10^11. But if the population has grown more rapidly, maybe 10^12. I think the more accurate estimate is 10^11, but I'm not certain. However, for the purposes of a Fermi estimate, maybe 10^11 is acceptable.

If the number of competitors is 10^3, then the fraction is 10^-8. But if the total number is 10^12, then 10^-9. But maybe the number of competitors is higher. For example, if there are multiple tournaments, maybe 10^4. But the question says "today’s tournament"—so it's one event. Let's stick with 10^3. So 10^-8.

But maybe the number of competitors is even higher. For example, if we consider that there are many tournaments happening today, maybe the total number of competitors is 10^5. But that's a stretch. Alternatively, maybe the number is in the hundreds of thousands. But that's more of a guess.

Alternatively, if the question is referring to a single tournament, like the World Cup, which has about 200-300 players, but that's a small number. But maybe the question is referring to all current competitors in all tournaments, which could be in the order of 10^5. But again, this is uncertain.

So, considering the uncertainty in the total number of humans (10^11 vs 10^12) and the number of competitors (10^3 vs 10^4), the fraction could be between 10^-8 and 10^-9. But also, if the number of competitors is even smaller, like 10^2, then it's 10^-9. So maybe the order of magnitude is 10^-9.

But wait, if the total number is 10^11, then 10^3 / 10^11 is 1e-8, which is 10^-8. If the total number is 10^12, then 1e-9. So depending on the total population, the exponent varies. Therefore, the answer could be in the range of -9 to -8. However, since the question is asking for the order of magnitude, which is the exponent, the answer is likely in the order of 10^-9, but maybe 10^-8. But I need to consider the uncertainty.

Alternatively, perhaps the number of competitors is even smaller. For example, if the tournament is a small local event, maybe 100 people. Then 100 / 10^11 is 1e-9. But if the tournament is large, like the Olympics, maybe 10^4, then 1e-7. But that's a huge range. So the answer depends on the two variables: total number of humans and number of competitors.

So to model this as a mixture of Gaussian components, considering different scenarios. For example, if the total humans is 10^11 and competitors is 10^3, then exponent is -8. If total is 10^12 and competitors is 10^3, exponent is -9. If competitors is 10^4, then with total 10^11, it's -7. But that's a lot of possibilities. But the question is to express the answer as a mixture of up to four Gaussian components. So I need to model possible scenarios.

Possible scenarios:

1. Total humans: 10^11, competitors: 10^3 → 1e-8.

2. Total humans: 10^12, competitors: 10^3 → 1e-9.

3. Total humans: 10^11, competitors: 10^4 → 1e-7.

4. Total humans: 10^12, competitors: 10^4 → 1e-8.

But that's four possibilities. However, these are not all equally likely. Maybe the main scenarios are:

- If the total humans is 10^11 and the number of competitors is 10^3 → -8.

- If the total humans is 10^12 and the number of competitors is 10^3 → -9.

But maybe also considering that the number of competitors could be in between. But for the purpose of the mixture, I need to model the possible exponents as a mixture.

So the possible exponents could be -9, -8, -7, etc. But given the uncertainty in both variables, the main components might be around -8 and -9. Let me think: if the total is 10^11 (100 billion) and the competitors are 10^3 (thousand), then 1e-8. If total is 10^12 (1 trillion) and competitors are 10^3, then 1e-9. If the total is 10^11 and competitors are 1e4, then 1e-7. But that's a higher exponent. However, given that the number of competitors is likely to be small, maybe the main components are around -8 and -9. Alternatively, maybe there's another component for lower exponents.

But since the question is about the order of magnitude, the answer is likely in the range of -9 to -8. However, I need to model this as a mixture of Gaussians. So perhaps two components: one centered

③ Gold label: y = -8.0000 (log10)

④ Model completion (answer after reasoning):

[no </think> — reasoning overran the token budget; no answer emitted]

Outcome: no answer block — <think> consumed the token budget → parse failure (reward floor).

Phase 3 — GRPO training on MoG-XML completed 2026-06-11

Results — held-out 200 Phase 2 cells, matched-subset (best checkpoint)

All numbers below are computed on the pairwise matched subset — the examples both the base and trained model parse — via matched_report.py. Parse rate is shown separately because the base model abstains (fails to parse) on the hard examples, so a parse-conditioned mean would flatter it (see the methodological lesson below).

dataset (ckpt)	parse rate		log-score mean ↑		log-score median ↑		abs err ↓		cov 90% (→0.9)
dataset (ckpt)	base	tr	base	tr	base	tr	base	tr	base	tr
ames (250) n=197	0.985	1.00	−0.707	−0.575	−0.493	−0.444	0.373	0.317	0.863	0.954
survival (250) n=143	0.715	1.00	−15.488	−2.106	−6.259	−1.875	2.734	2.326	0.343	0.804
fermieval (300) n=132	0.660	1.00	−6.333	−3.877	−1.172	−1.549	6.957	3.695	0.780	0.818

fermieval nuance: training improves the matched mean (−6.33 → −3.88, far fewer catastrophic tail misses, abs-err 6.96 → 3.70) but the median slips (−1.17 → −1.55) — mild over-sharpening on the typical question while the tails tighten. The other two datasets improve on every metric.

Phase 3 verdict. GRPO with the clipped log-score reward improves the trained format on all three datasets (matched-subset comparison), takes parse rates to 1.00 everywhere, and moves coverage toward nominal (survival 0.34 → 0.83). Training dynamics showed a recurring self-correction: runs twice dipped by over-sharpening (σ down, tail scores collapse) and recovered by re-widening and adding mixture components — the proper scoring rule disciplining overconfidence in real time. Checkpoint selection matters (ames and survival peaked at 250, fermieval at 300).

Methodological lesson (applies to every table in this project). Parse-conditioned log-score means are not comparable across models or methods with different parse rates: parse failure is selective abstention on hard examples (the same commit-vs-abstain behavior the forecasting prompt-ablation found for Beta). Compare on matched subsets, or impute failures explicitly.

Example rollouts — trained MoG-XML (click to expand)

ames_housing · trained MoG-XML — parse rate 1.00, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating the sale price of a residential property in Ames, Iowa (sales 2006-2010).

Property features:
- Neighborhood: Brookside
- MS_Zoning: Residential_Medium_Density
- House_Style: One_and_Half_Fin
- Year_Built: 1939
- Year_Remod_Add: 1950
- Overall_Qual: Above_Average
- Overall_Cond: Above_Average
- Gr_Liv_Area: 1412
- Total_Bsmt_SF: 780
- Garage_Type: Detchd
- Garage_Cars: 1
- Full_Bath: 1
- Half_Bath: 0
- Bedroom_AbvGr: 4
- Pool_QC: No_Pool
- Fireplaces: 1
- Lot_Area: 6000
- Central_Air: Y

Provide your estimate of the natural log of the sale price in USD (typical values 10.5 to 13.5).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the sale price for this house in Ames, Iowa from 2006-2010. Let me start by looking at the property features. The neighborhood is Brookside, which is a good area. The MS_Zoning is Residential_Medium_Density, so that's a medium density zoning, which might affect the price. The house style is One_and_Half_Fin, which is a common style, maybe a bit older. The house was built in 1939 but remodeled in 1950, so it's been updated. Overall Qual and Condition are Above Average, which is positive. Gr_Liv_Area is 1412 sqft, and Total Bsmt is 780, so that's a good size. Garage is detached with 1 car space. One full bath, no half baths. Four bedrooms. No pool. One fireplace. Lot area is 6000 sqft, which is decent. Central air is yes.

Now, I need to think about how these factors influence the price. The location (Brookside) is likely a premium. The house is well-maintained with above average quality. The size is good, but the year built is 1939, which might be a bit old, but the remodel in 1950 could have improved it. Central air is a plus. The lot size is good.

I need to consider typical values for the natural log of sale prices, which are between 10.5 and 13.5. Let me think about possible components. Maybe a central component around 12.0, considering the above-average features. But there's uncertainty. If the house is in a good neighborhood but has an older structure, maybe a lower value? Or if the central air and size are strong points, higher. Maybe two components: one around 11.5 with higher sigma for uncertainty, and another around 12.5. Wait, but the user said up to 4 components. Maybe more. But I need to balance. Let me check if there are factors that could push it higher or lower. The lot size is large, which might be a plus. The central air is a modern feature. The overall quality is good. So maybe a central component at 12.0 with a moderate sigma, and another component at 12.5 with a smaller weight. Or maybe a third component if there's more uncertainty. But I think two components would be enough. Let me go with two components: one around 11.8 with a sigma of 0.3, and another around 12.2 with a sigma of 0.3. Weights would be around 0.6 and 0.4. That way, it's a mix between lower and higher estimates, reflecting uncertainty in different scenarios.

③ Gold label: y = 11.7943 (log USD)

④ Model completion (answer after reasoning):

<mixture>
  <component><mu>11.8</mu><sigma>0.3</sigma><w>0.6</w></component>
  <component><mu>12.2</sigma>0.3</sigma><w>0.4</w></component>
</mixture>

Parsed → induced: 1.00·N(11.80, 0.30); log-score 0.285

survival · trained MoG-XML — parse rate 1.00, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating a time-to-event outcome in the gbsg survival cohort. The feature columns are anonymized; reason from the marginal distribution and any plausible scale you can infer from the cohort name.

Patient features:
  x0: 1
  x1: 0
  x2: 1
  x3: 57
  x4: 1
  x5: 4
  x6: 271

Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, let's tackle this. The user is asking for an estimate of the natural log of time to event in the GBSG cohort. The features given are anonymized, so I need to infer their possible meanings based on the cohort name. GBSG is the German Breast Cancer Study Group, which focuses on breast cancer survival. Time-to-event outcomes in such studies are often in months or years.

Looking at the feature values: x3 is 57, which might be age (common in cohorts). x6 is 271, which could be a survival time in months. The other features are 0s and 1s, maybe binary variables like treatment type. Since the question is about the log of time, if the actual time is in months, the log would be in ln(months). 

For the mixture of Gaussians, I need to consider possible distributions. If the survival time is in months, the log might have a mean around 20-30 (if the event is within a few years), with a standard deviation of 10-20. But since GBSG is a breast cancer study, survival times can vary. Maybe there's a component for shorter times (e.g., 10-20 months) and a longer tail (e.g., 30-50 months). Also, considering possible censoring, the distribution might have a right skew, so a mixture with two components: one for early events and one for longer survival. Weights could be 0.5 each. Let me check: if the features are related to treatment, maybe the log time has a mean of 25 with SD 15, and another component at 40 with SD 20. That seems plausible.

③ Gold label: y = 3.3307 (log time) [right-censored]

④ Model completion (answer after reasoning):

<mixture>
  <component><mu>25</mu><sigma>15</sigma><w>0.4</w></component>
  <component><mu>40</mu><sigma>20</sigma><w>0.6</w></component>
</mixture>

Parsed → induced: 0.40·N(25.00, 15.00) + 0.60·N(40.00, 20.00); log-score -0.051

fermieval · trained MoG-XML — parse rate 1.00, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: 2^32

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a mixture of up to 4 Gaussian components. Use multiple components when the answer is genuinely uncertain across qualitatively different scenarios. Output exactly:
<mixture>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
  <component><mu>MU</mu><sigma>SIGMA</sigma><w>WEIGHT</w></component>
</mixture>
All values numeric; weights need not sum to 1 (we normalize).

② Model reasoning (<think>):

Okay, I need to estimate 2^32 in terms of a base-10 exponent. Let me recall that 2^10 is about 1000, so 2^30 is roughly 10^9. Then 2^32 would be 4 times that, so around 4*10^9. So the order of magnitude is 10^9. But wait, maybe I should check more precisely. 2^10 is 1024, which is about 10^3. So 2^30 is (2^10)^3 ≈ (10^3)^3 = 10^9. Then 2^32 is 2^30 * 2^2 ≈ 10^9 * 4 = 4*10^9. So the base-10 exponent is 9. But maybe there's some uncertainty here. If I consider that 2^10 is slightly more than 10^3, like 1.024*10^3, then 2^30 would be (1.024)^3 *10^9 ≈ 1.0737 *10^9. Then 2^32 is 4 * 1.0737 *10^9 ≈ 4.2948*10^9. So the exponent is still around 9. So the order of magnitude is 10^9. Therefore, the answer is 9. But maybe there's some uncertainty in the approximation. So I can model this with a Gaussian around 9 with a small sigma, maybe 0.5. But if I'm not sure about the exact value, maybe I should have a mixture of components. For example, if I think it's 9 with high probability, but maybe 8 or 10 as alternatives. So the mixture could be 9 with a sigma of 0.5, and maybe 8 and 10 with smaller weights. But the main component is 9. So the final answer is a mixture with a peak at 9, and maybe a small spread.

③ Gold label: y = 9.0000 (log10)

④ Model completion (answer after reasoning):

<mixture>
  <component><mu>9</mu><sigma>0.5</sigma><w>0.8</w></component>
  <component><mu>8</mu><sigma>1</sigma><w>0.1</w></component>
  <component><mu>10</mu><sigma>1</sigma><w>0.1</w></component>
</mixture>

Parsed → induced: 0.80·N(9.00, 0.50) + 0.10·N(8.00, 1.00) + 0.10·N(10.00, 1.00); log-score -0.376

Design

Reward: $ R = \mathrm{clip}(\log p_{\text{MoG}}(y^{*}),\, -50,\, +10) $ on parse success; parse failure = −55 (strictly below the floor, so abandoning the format is never optimal — the DDXPlus/HotpotQA lesson). Censored survival rows score $\log S(y)$. The clip floor bounds catastrophic-outlier gradients (one −10⁵ log-density would dominate a batch) while preserving direction. The reward calls the same parse/density code as the Phase 2 eval, so reward and metric are identical by construction.
Config: LoRA r=64, lr 1e-5 cosine, 300 steps/dataset, group 16, effective batch 512, vLLM colocate; checkpoints every 25 steps, resume-safe (already survived one crash and one power loss today).
Split protocol: the first 200 examples of the fixed seed-0 shuffle — the exact Phase 2 eval cells — are never trained on; during-training eval uses the first 100 of them every 50 steps. Post-training: merge LoRA, run the full 200-cell Phase 2 eval, compare against the baseline tables above (e.g. ames mog_xml base = −0.707).

Phase 4 — pluck programs: GRPO trained in progress 2026-06-18

The model emits a Pluck program; exact inference induces its density, scored by the same clipped log-score reward as Phase 3 (reward_pluck.py caches each inference so the reward + 5 metric heads cost one program execution, ~1 ms/rollout). Training order survival → fermieval → ames, 300 steps each, LoRA. Two things resolved a prior open question:

No format curriculum needed. GRPO lifts the pluck parse rate on its own — survival 0.70 → 1.00 by step ~26, exactly like MoG-XML. An improved-prompt variant (pluck_v2: explicit no-division / binary-+. rules + a log-space recipe) was tested zero-shot and hurt (survival parse 0.70 → 0.33; the denser prompt worsened <think> overrun), so it was dropped — the base model needs fewer rules, not more; GRPO is what fixes the language gaps.
Trained pluck ≈ a MoG expressed as a program. Reading the rollouts (not just the metric): the model branches with flip/if, but between two gaussians, inducing a gaussian mixture (≈1.8 components). It does not reach for point-mass "cure" leaves — the one structure a plain MoG cannot express.

Trained pluck vs trained MoG, matched subset (best checkpoint each)

The fair comparison is trained-vs-trained on the full held-out 200 cells (both parse everything). Base baselines (parse-conditioned) shown for context. All three datasets done.

dataset · series	parse	log-score mean ↑	median ↑	abs err ↓	CRPS ↓	cov 90%
survival · base MoG / base pluck	.72/.70	−15.5/−6.7	−6.3/−2.3	—	—	.34/.51
survival · trained MoG (s250)	1.00	−1.99	−1.79	2.61	1.71	0.83
survival · trained pluck (s250)	1.00	−2.10	−1.79	1.79	1.25	0.81
fermieval · base MoG / base pluck	.66/.47	−6.3/−15.2	−1.2/−1.4	—	—	.75/.58
fermieval · trained MoG (s300)	1.00	−7.11	−1.62	4.20	3.52	0.78
fermieval · trained pluck (s250)	1.00	−4.93	−1.61	3.26	2.80	0.80
ames · base MoG / base scalar	.99/.91	−0.71/−1.15	−0.49/−0.69	—	—	.86/.69
ames · trained MoG (s250)	1.00	−0.567	−0.440	0.314	0.229	0.955
ames · trained pluck (s250)	1.00	−0.555	−0.421	0.304	0.220	0.950

Survival verdict: a trade-off, not a winner. Trained MoG and trained pluck tie on the log-score median (−1.79) and sit within 0.1 nat on the mean (MoG marginally ahead on tail log-density). But pluck is clearly sharper on point accuracy and spread: abs-err −31% (1.79 vs 2.61) and CRPS −27% (1.25 vs 1.71), at the same ~0.8 coverage. Exact inference seems to yield cleaner means than directly emitting XML component parameters. Both turn the catastrophic base baselines (−6.66 / −15.49) into ≈−2.

Structure probe: GRPO leaves pluck's one free lunch on the table. Splitting survival by censoring: trained pluck actually wins on the 75 censored rows (−1.40 vs MoG −1.49) and loses on the 125 uncensored (−2.52 vs −2.29). But here is the finding — on a censored patient a point mass at $T=\infty$ ("cured, no event") is exactly right, and a gaussian MoG provably cannot express it, yet trained pluck never emits a point/label leaf (mass 0.000, max 0.00 even on censored rows). It converges entirely to the gaussian-mixture subset that MoG already covers. So the PPL's expressivity advantage is latent but unrealized under the log-density reward — a wide gaussian gives a decent $\log S(t)$, GRPO rarely samples the point-mass branch, and the few-shot exemplar uses gaussians. This sharpens the paper's framing: richer output language ≠ richer learned structure unless the reward or exploration pushes for it.

fermieval verdict: pluck wins — but not via structure. On the full held-out 200 cells, trained pluck beats trained MoG: mean log-score −4.93 vs −7.11 (+2.2 nats), abs-err 3.26 vs 4.20, CRPS 2.80 vs 3.52, coverage 0.80 vs 0.78; median tied (−1.61). The win is on the hard tail — on the easy n=67 subset where all methods parse, MoG is marginally ahead (−2.40 vs −2.51). Mechanism: fermieval pluck collapses to a single wide gaussian over the exponent (0/32 programs branch; ncomp≈1.0), and that wide gaussian is adaptive for the heavy-tailed Fermi answers — it avoids the catastrophic clips MoG's mixture hits when a hard question is orders of magnitude off. The hypothesized (+.) log-space factor composition is never used; the model directly estimates the final exponent.

Headline (all 3 datasets): trained pluck ties-or-beats trained MoG — as an inference engine, not via richer structure. Trained pluck wins or ties trained MoG on every dataset: ames narrowly (−0.555 vs −0.567, all metrics), survival a trade-off (MoG +0.1 nat mean, pluck −31% abs-err / −27% CRPS), fermieval a clear win (+2.2 nats). Biggest gains on the heavy-tailed case. Yet in no case does pluck use the structure only a program can express — point masses (survival), log-space factor composition (fermieval): flip/ if just produces gaussian mixtures, or collapses to a single gaussian. The payoff is cleaner means and adaptive spread from exact inference over a clean parametric family, not the latent expressivity. That reframes the PPL pitch: today it's a better-conditioned way to emit a mixture; realizing structural advantages needs a reward or exploration signal that pays for structure (the cure-exemplar probe below tests the exploration side).

Follow-up: cure-exemplar probe — does showing structure unlock it?

A survival run (--method pluck_cure) adds one worked exemplar showing a point-mass cure branch ((if no-event 99.0 (gaussian 4.0 1.5))), everything else identical to v1, to separate an exploration/prompt limit from a reward limit. The result is nuanced — and surprising.

survival rows	n	trained MoG	v1 pluck	cure pluck	cure point-mass	% rows w/ point-mass
censored	74	−1.481	−1.399	−1.411	0.015	5%
uncensored	125	−2.294	−2.516	−2.115	0.028	11%
all	199	−1.992	−2.101	−1.853	0.023	9%

Showing structure helps the model — but not by using the structure correctly. The cure exemplar produces the best survival model of all (−1.853, beating both trained MoG −1.99 and v1 pluck −2.10, with better coverage 0.88–0.91). But the gain is entirely on uncensored rows (−2.12 vs v1 −2.52); on censored rows it's a wash (−1.41 ≈ v1 −1.40). And point masses are used more on uncensored rows (2.8%) than censored (1.5%) — the opposite of where a cure helps. So the answer is "both, neither fully": the structure is reachable via prompting (the model does emit point masses, vs v1's exact 0), not a pure exploration limit — but the reward/credit-assignment never teaches the model to place point masses where they belong (conditioned on the censoring signal), so it doesn't realize the structure's intended benefit. The overall improvement comes from a generally better-shaped prior, not correct structural use.

Variance control (2 runs each). v1 pluck across two runs lands at −2.10 and −2.17/−2.22; the cure model across two runs lands at −1.85 and −1.92. Both cure runs beat both v1 runs — the distributions don't overlap, so the ~0.27 nat cure gain is real and reproducible (cure ~−1.90 > trained MoG −1.99 > v1 pluck ~−2.16). But all runs share a median ~−1.70…−1.79, so the gain lives in the mean (tail robustness, coverage 0.86–0.91), not the typical case — a better-calibrated prior, not correct deployment of the cure structure.

Phase 4 conclusion. A richer output language (a probabilistic program with exact inference) buys better-conditioned distributions essentially for free — trained pluck ties-or-beats trained MoG-XML on every dataset, and seeding structure yields a real extra gain. But across E1–E4 the model never realizes the language's distinctive structural expressivity (point masses, factor composition): the plain log-density reward doesn't pay for it, and even when a prompt makes the structure reachable, credit assignment doesn't teach correct placement. Realizing PPL structure needs a reward or exploration signal that rewards structure — that's the sharpened direction for the PPL arc. (As a no-PPL intermediate, MoG-XML remains the workhorse; pluck's exact inference is the value-add today.)

Example rollouts — pluck programs (click to expand)

The failure traces below show the two distinct ways a pluck rollout fails: a program is emitted but the language rejects it (variadic +., division, scientific notation), or <think> overruns the budget so no program appears at all.

survival · base pluck — parse rate 0.70, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating a time-to-event outcome in the gbsg survival cohort. The feature columns are anonymized; reason from the marginal distribution and any plausible scale you can infer from the cohort name.

Patient features:
  x0: 0
  x1: 1
  x2: 1
  x3: 55
  x4: 2
  x5: 26
  x6: 53

Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the time to event for the GBSG cohort. The features given are x0 to x6, but they're anonymized. The cohort name is GBSG, which I know is the German Breast Cancer Study Group. From what I remember, GBSG data is often in months, but I should confirm that. Wait, the user mentioned that for GBSG, the time unit is months. But the question says to provide the answer in the same time unit as the cohort. Wait, the user says "Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG)." So for GBSG, it's months. So the answer should be in months, but the log of that time.

But how to model this? The features are x0 to x6, which are 0,1,1,55,2,26,53. But they are anonymized, so I can't know their actual meaning. However, maybe I can infer their scale based on GBSG data. For example, in GBSG, common features might include things like age, tumor size, etc. But since the features are anonymized, I have to make assumptions.

Alternatively, maybe the features are numerical values that are part of the model. But without knowing their actual meaning, perhaps I can assume that the log of the time to event is a normal distribution with some mean and standard deviation. But how to estimate that?

Alternatively, maybe the features are part of a model where each feature contributes to the log-time. But since the user says to reason from the marginal distribution and any plausible scale, perhaps I can assume that the log-time is normally distributed with a mean and standard deviation based on typical GBSG data.

Wait, but the problem says to express the belief as a probabilistic program. So I need to model uncertainty. Let's think: the log of the time to event. For GBSG, the time is in months. Suppose the log of the time is normally distributed. But how to estimate the mean and standard deviation?

Alternatively, maybe the features are part of a model that affects the log-time. For example, if some features are binary (like x0=0, x1=1, x2=1), and others are continuous (like x3=55, x4=2, x5=26, x6=53). But without knowing the actual variables, perhaps I can assume that the log-time is a combination of these features. However, the user might expect me to make a simple model.

Alternatively, maybe the features are not directly related to the log-time, but the problem is to estimate the log-time based on the cohort's typical values. For example, if the GBSG data has a typical survival time of, say, 10 months, then log(10) is about 2.3. But I need to think about the scale.

Wait, but the user says to reason from the marginal distribution and any plausible scale. So perhaps I can assume that the log-time is normally distributed with a mean of 5.0 and standard deviation of 1.0, but that's arbitrary. Alternatively, maybe the log-time is around 5 months, so log(5) is about 1.6, but that's not right. Wait, no, the log is of the time in months. So if the time is in months, then the log is the natural logarithm of the number of months.

But without knowing the actual distribution, perhaps the best approach is to model the log-time as a normal distribution with some mean and standard deviation. Since the problem allows for a probabilistic program, I can use a Gaussian distribution with some parameters.

But how to choose the mean and standard deviation? If I have no specific information, maybe I can use a mean of 5.0 and standard deviation of 1.0. But I need to make sure that the program reflects my uncertainty. Also, the problem says to use flip+if branches when there are different scenarios. For example, if there's a rare scenario where the log-time is higher, I could model that.

Alternatively, maybe the features are part of a model that affects the log-time. For example, if some features are binary (like x0=0, x1=1, x2=1), maybe they are indicators for certain factors. But without knowing the actual variables, this is speculative. However, the problem says to reason from the marginal distribution and any plausible scale. So perhaps the log-time is a normal distribution with a mean of 5.0 and standard deviation of 1.0, and there's a rare scenario where it's higher, like 9.0 with 0.2 probability.

So the program could be something like:

(let ((typical (gaussian 5.0 1.0))
(rare (flip 0.2)))
(if rare (gaussian 9.0 2.0) typical))

This would model the log-time as a Gaussian with mean 5 and SD 1, but with a 20% chance of being a higher Gaussian with mean 9 and SD 2. That's a simple way to express uncertainty. The mean and SD are chosen based on typical GBSG data, but I'm not sure. Alternatively, maybe the mean is 5.0 and SD 1.0, but I need to check if that's plausible.

Alternatively, maybe the log-time is around 5 months. So log(5) is about 1.6, but that's not the case. Wait, no. The time is in months, so the log of the time (in months) would be log(months). For example, if the time is 5 months, log(5) is ~1.6. But if the time is 10 months, log(10) is ~2.3. So if the typical log-time is around 5, that would mean the time is e^5 ≈ 148 months, which is about 12 years. That seems too long for GBSG data. Wait, GBSG is a breast cancer study, so survival times are usually in months, but maybe up to a few years. So 10-20 months is more typical. So log(10) is ~2.3, log(20) is ~3.0. So maybe the mean is around 3.0 and SD 1.0. But I'm not sure. Alternatively, maybe the log-time is around 5.0, but that would imply a time of e^5 ≈ 148 months, which is 12.3 years. That seems too long. So perhaps the mean is 3.0, SD 1.0. But again, without knowing the actual data, this is just a guess.

Alternatively, maybe the features are not directly related, and the log-time is a simple Gaussian. Since the problem allows for a probabilistic program, I can choose parameters that reflect my uncertainty. So maybe the typical case is a Gaussian with mean 5.0 and SD 1.0, and a rare case of 9.0 with SD 2.0. That's a possible approach.

So the final answer would be a program that models this. The user wants the answer in the same time unit as the cohort, which is months for GBSG. Therefore, the log of the time is in natural logarithm of months. So the program would generate a value based on the Gaussian parameters. The answer would be the log of the time, so the program's output is the log of the time in months.

Therefore, the program is as I wrote before: a typical Gaussian with mean 5.0 and SD 1.0, with a 20% chance of being a higher Gaussian. This reflects uncertainty, with a rare scenario where the log-time is higher.

③ Gold label: y = 3.7914 (log time) [right-censored]

④ Model completion (answer after reasoning):

<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

Parsed → induced: 0.80·N(5.00, 1.00) + 0.20·N(9.00, 2.00); log-score -0.096

✗ Rejected — program/mixture emitted but rejected

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating a time-to-event outcome in the support survival cohort. The feature columns are anonymized; reason from the marginal distribution and any plausible scale you can infer from the cohort name.

Patient features:
  x0: 71.49
  x1: 1
  x2: 4
  x3: 1
  x4: 1
  x5: 1
  x6: 2
  x7: 146
  x8: 152
  x9: 44
  x10: 38.8
  x11: 150
  x12: 29.9
  x13: 1.8

Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the time to event for the SUPPORT cohort. The features given are anonymized, so I have to infer their meanings based on the cohort name and typical features in survival analysis.

First, the SUPPORT cohort is known for patients in the ICU, so features might relate to things like age, blood pressure, heart rate, etc. Let's look at the numerical values. x0 is 71.49, which could be age. x1 to x5 are 1s, maybe indicating presence or absence of certain conditions. x6 is 2, perhaps a count of something. x7 is 146, maybe a blood pressure value. x8 is 152, maybe another vital sign. x9 is 44, maybe a lab result. x10 is 38.8, maybe a blood glucose level. x11 is 150, maybe a creatinine level. x12 is 29.9, maybe a white blood cell count. x13 is 1.8, maybe a score like the SOFA score.

Since the question is about the natural log of time to event, I need to think about what factors influence survival in SUPPORT. Typically, features like age, blood pressure, and other vital signs might be important. But without knowing the exact features, I have to make educated guesses.

Assuming that the log-time is influenced by some of these variables. For example, age might have a negative effect (older patients have worse outcomes), so maybe a negative contribution from x0. Blood pressure (x7) might be positive if higher BP is associated with longer survival, but that's not always the case. Alternatively, maybe some features are more important than others.

But given that the features are anonymized, and the cohort is SUPPORT, which is a real-world dataset, I should recall that in the SUPPORT study, the primary outcome was 28-day mortality, and features included things like age, systolic blood pressure, heart rate, etc. However, the exact variables here are not known, so I have to make assumptions.

The problem says to reason from the marginal distribution and plausible scale. Since the answer is the natural log of time to event, which is typically modeled with a log-normal distribution, but here we are to estimate the mean of the log-time.

If I consider that the log-time is a combination of several features, each contributing some uncertainty. For example, age (x0) might have a negative effect, so maybe a term like (gaussian -1.0 0.5). Blood pressure (x7) could be a positive term, say (gaussian 0.5 0.3). Other variables might contribute smaller effects.

But since the problem allows for a probabilistic program, I need to model the uncertainty. Maybe some features are more important than others. For instance, age is a major factor. If x0 is age, then a negative term. Other features like blood pressure (x7) might be positive, but with lower certainty.

Alternatively, maybe the log-time is a combination of several Gaussian terms. For example, if age is a major factor, and other variables contribute smaller effects.

But the user wants a probabilistic program that reflects uncertainty. So perhaps the log-time is a sum of several Gaussian variables, each representing the contribution of a feature. For example, age (x0) might contribute a negative value, and other features contribute small positive or negative values.

But since the exact features are unknown, I need to make plausible guesses. Let's assume that the log-time is a combination of several Gaussian terms. For instance, age (x0) could contribute a term with mean -1.0 and standard deviation 0.5. Other features like blood pressure (x7) could contribute a term with mean 0.5 and SD 0.3. Maybe other features have smaller contributions, like 0.1 or -0.2.

Alternatively, if some features are binary (like x1 to x5 being 1s), maybe they have a small effect. For example, if x1 is a binary variable indicating presence of a condition, it might add a small value, but with low probability.

But since the problem says to use flip+if branches when different scenarios are plausible, maybe some features are more uncertain. For example, if x7 is blood pressure, maybe it's a positive term with some uncertainty.

Putting it all together, I might model the log-time as a sum of several Gaussian terms. For example, a main term from age, plus some other terms from other features. Since the exact features are unknown, I'll have to make plausible assumptions.

Let me think: the log-time is the natural log of the time to event. In survival analysis, the hazard function is often modeled, but here we are to estimate the log-time. Assuming that the log-time is a linear combination of features, each with their own mean and variance.

But without knowing the exact features, I have to make assumptions. Let's say that the log-time is a Gaussian with mean around 0.0 and standard deviation 1.0, but that's too simplistic. Alternatively, considering that in the SUPPORT cohort, the average time to event (28 days) would have a log of about ln(28) ≈ 3.33. But the actual log-time would depend on the model.

Alternatively, if the features are such that the log-time is influenced by several variables, each contributing a small amount. For example, the main term could be a Gaussian with mean around 0.0 and SD 1.0, and then add some other terms.

But maybe the features are such that the log-time is a combination of several independent factors. For example, age (x0) contributes a negative term, blood pressure (x7) contributes a positive term, and other features contribute smaller terms.

Putting this into a probabilistic program, perhaps the log-time is a sum of several Gaussians. For example:

(let ((age-term (gaussian -1.0 0.5))
(bp-term (gaussian 0.5 0.3))
(others (gaussian 0.0 0.2)))
(+. age-term bp-term others))

But this is just a rough guess. Alternatively, if some features are more uncertain, maybe some terms are flipped with certain probabilities. For instance, if a feature is binary, maybe it's a flip with a certain probability.

But given the lack of information, I think the best approach is to model the log-time as a Gaussian with a mean of 0.0 and standard deviation 1.0, but that's probably not accurate. However, considering that the actual log-time in the SUPPORT cohort is around ln(28) ≈ 3.33, but the model might have different parameters.

Alternatively, if the log-time is a sum of several terms, each with their own mean and variance. For example, age (x0) contributing -1.0, blood pressure (x7) +0.5, and other terms adding up to 0.0.

But since the user wants a probabilistic program that reflects uncertainty, maybe some terms are added with uncertainty. For example:

(let ((age (gaussian -1.0 0.5))
(bp (gaussian 0.5 0.3))
(others (gaussian 0.0 0.2)))
(+. age bp others))

This would be a simple model where age and blood pressure contribute, and others contribute a small amount.

Alternatively, if some features are more uncertain, maybe a flip with a certain probability. For example, if a feature is more uncertain, like a rare event, then use a flip.

But given that the problem allows for flip+if branches, maybe some features are more uncertain. For example, if x1 is a binary variable, maybe it's a flip with a certain probability.

But given that the exact features are unknown, I think the best approach is to make a simple model with a Gaussian mean and standard deviation, considering that the log-time is influenced by several factors.

So the final answer would be a program that models the log-time as a Gaussian with mean 0.0 and standard deviation 1.0, but adjusted based on plausible features. However, since the actual answer is not known, I have to make an educated guess.

Alternatively, maybe the log-time is around 0.0, but that might not be correct. However, given the lack of information, I think the best approach is to use a Gaussian with mean 0.0 and standard deviation 1.0, as a starting point.

③ Gold label: y = 6.1944 (log time) [right-censored]

④ Model completion (answer after reasoning):

<program>
(let ((age (gaussian -1.0 0.5))
      (bp (gaussian 0.5 0.3))
      (others (gaussian 0.0 0.2)))
  (+. age bp others))
</program>

Outcome: answer emitted but the scorer rejected it — PanicException: assertion `left == right` failed: FAdd expected 2 arguments, got 3 left: 3 right: 2 → counted as a parse failure (reward floor).

✗ Overrun — reasoning overran, no answer

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating a time-to-event outcome in the support survival cohort. The feature columns are anonymized; reason from the marginal distribution and any plausible scale you can infer from the cohort name.

Patient features:
  x0: 50.47
  x1: 1
  x2: 0
  x3: 1
  x4: 0
  x5: 0
  x6: 1
  x7: 120
  x8: 124
  x9: 32
  x10: 39.09
  x11: 142
  x12: 14.8
  x13: 0.7

Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the time to event for the SUPPORT cohort. The features given are a list of numbers, but they're anonymized. The cohort name is SUPPORT, which I remember is a study that looked at patients in the ICU, so time to event might be in days.

First, I should think about what these feature columns might represent. Since the features are anonymized, I can't know exactly what they are, but maybe I can infer their possible scales based on typical medical features. For example, x0 is 50.47, which could be age (in years?), but that's just a guess. x1 is 1, x2 is 0, etc. Maybe some are binary indicators (like 0 or 1) for conditions or treatments. x7 is 120, x8 is 124, maybe blood pressure or some lab values. x9 is 32, maybe a count of something. x10 is 39.09, which could be a lab value like creatinine. x11 is 142, maybe a blood pressure reading. x12 is 14.8, maybe a blood glucose level. x13 is 0.7, maybe a small value like a fraction.

But since the question is about the natural log of time to event, I need to think about how these features might influence that. However, without knowing the actual variables, I have to make assumptions. The problem says to reason from the marginal distribution and any plausible scale from the cohort name. Since the cohort is SUPPORT, which is about ICU patients, the time to event is likely in days.

Assuming that the features are independent, I can model each as a Gaussian with some mean and standard deviation. But how do I choose the parameters? For example, if a feature is a binary variable (like x1=1, x2=0), maybe it's a dummy variable indicating presence of a condition. But since they are 0 or 1, maybe they are not continuous. However, the problem says to use Gaussian, so perhaps I need to model each as a Gaussian with some mean and std dev.

Alternatively, maybe the features are part of a model where their effect on the log-time is additive. For instance, each feature might contribute a certain amount to the log-time. But since the features are anonymized, I need to make educated guesses about their possible contributions.

Alternatively, maybe the features are part of a model where the log-time is a sum of some terms. For example, if x0 is age, then maybe it's a linear term. But without knowing the actual variables, this is difficult. However, the problem says to use the marginal distribution and plausible scales.

Alternatively, maybe the features are such that the log-time is a combination of some known variables. For example, in the SUPPORT study, variables like age, blood pressure, creatinine, etc., might be factors. But again, without knowing the exact variables, I have to make assumptions.

But the problem says to use the marginal distribution. So perhaps each feature is a Gaussian with a certain mean and SD. For example, if x0 is age (in years?), then maybe it's around 50, with SD 10. But the value given is 50.47, which is close to 50. So maybe x0 is age, and the mean is 50.47, but that's just one data point. However, since the features are anonymized, maybe each feature is a Gaussian with some mean and SD that I can infer based on typical values.

Alternatively, maybe the log-time is a combination of the features. For example, if x7 is 120, maybe that's a blood pressure value, and x8 is 124, which is another value. Maybe the log-time is affected by these values. But again, without knowing the actual variables, this is speculative.

Alternatively, maybe the log-time is a linear combination of the features. For example, if x0 is age, x1 is a binary variable (like 1 for male), x2 is another binary variable (like 0 for no comorbidity), etc. But again, without knowing the actual variables, I need to make assumptions.

Since the problem requires a probabilistic program, I need to model the log-time as a Gaussian with some mean and SD. But how to determine those parameters? Maybe I can use the values given for each feature to estimate their mean and SD. For example, if x0 is 50.47, maybe that's the mean for that feature. But that's just a single value. However, the problem says to reason from the marginal distribution, so perhaps each feature is modeled as a Gaussian with a certain mean and SD based on the data.

Alternatively, maybe the log-time is a sum of the features, each contributing a certain amount. For example, if each feature is a Gaussian, then the total log-time is the sum of those Gaussians. But how to determine the parameters?

Alternatively, the log-time could be a single Gaussian, with a mean and SD that I can estimate based on the cohort. For example, if the SUPPORT study has a typical log-time of 20 days, then the mean could be 20, and SD 5. But that's just a guess.

Alternatively, given that the features are part of the model, maybe the log-time is a function of the features. For example, if x0 is age, x1 is a binary variable, etc., then the log-time could be a linear combination of these features. But again, without knowing the actual variables, this is hard.

But the problem says to use the marginal distribution and plausible scales. So perhaps each feature is a Gaussian with a certain mean and SD. For example, x0 is 50.47, so maybe that's the mean of x0. But since it's a single value, that's not helpful. However, the problem says to use the marginal distribution, so maybe I can assume that each feature is a Gaussian with a mean and SD that are typical for the variables in the cohort.

Alternatively, perhaps the log-time is a single Gaussian with mean 5.0 and SD 1.0. But that's arbitrary.

Wait, the problem says to express the belief as a probabilistic program. The program should reflect my uncertainty. So I need to model the log-time as a Gaussian with some mean and SD, but also consider possible scenarios where different features might affect it.

But given that the features are anonymized, and the cohort is SUPPORT, which is about ICU patients, and the time to event is in days, maybe I can assume that the log-time is around 5 days (since log(5) is about 1.6, but that's not helpful). Alternatively, if the time is in days, the log-time would be log(days). For example, if the time is 10 days, log(10) is about 2.3. But without knowing the actual time, this is guesswork.

Alternatively, maybe the log-time is a sum of the features. For example, each feature contributes a certain amount. So if x0 is 50.47, maybe that's a feature that contributes 50.47 to the log-time. But that seems unlikely, as log-time would be in days, and 50 would be too large. Wait, no, the log-time is the natural log of the time, so if the time is in days, then log(time) would be log(days). For example, if the time is 10 days, then log(10) is ~2.3. If the time is 100 days, log(100) is ~4.6.

But how to get this from the features? The features given are numbers like 50.47, 1, 0, etc. If the features are part of a model where the log-time is a combination of them, perhaps each feature contributes a certain amount. For example, maybe the log-time is the sum of some features multiplied by their coefficients.

But without knowing the actual variables, I can't be precise. However, the problem allows using Gaussian distributions for each feature. So maybe each feature is a Gaussian with a certain mean and SD. For example, x0 is 50.47, so maybe it's a Gaussian with mean 50.47 and SD 10. Then, the log-time is a sum of these features. But again, this is speculative.

Alternatively, maybe the log-time is a single Gaussian with mean 5.0 and SD 1.0. But that's arbitrary.

Alternatively, since the features are given as numbers, perhaps the log-time is the sum of the features. But that's not possible because the features are in different scales. For example, x0 is 50, x1 is 1, x2 is 0, etc. Adding them would give a very large number, which would be impossible for log(time) in days. So that's probably not the case.

Alternatively, maybe the log-time is a combination of some features. For example, if x0 is age, x7 is blood pressure, x8 is another value, etc. But again, without knowing the actual variables, this is guesswork.

Given that the problem requires a probabilistic program, perhaps the best approach is to assume that the log-time is a Gaussian with a certain mean and SD, based on the cohort. For example, in the SUPPORT study, the time to event (like mortality) is often in days, and the log-time might be around 5 days (log(5) ~ 1.6), but that's

③ Gold label: y = 7.5055 (log time) [right-censored]

④ Model completion (answer after reasoning):

[no </think> — reasoning overran the token budget; no answer emitted]

Outcome: no answer block — <think> consumed the token budget → parse failure (reward floor).

fermieval · base pluck — parse rate 0.47, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: On average, how many kilometers per second would Santa need to travel to go around the circumference of the earth once in a 24-hour period?

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, let's tackle this Fermi question. The user wants to estimate how many kilometers per second Santa needs to travel to go around the Earth once in 24 hours. First, I need to figure out the circumference of the Earth. I remember the Earth's radius is roughly 6,371 km, so the circumference would be 2 * π * radius. Let me calculate that: 2 * π * 6371 ≈ 40,074 km. That's the distance Santa has to cover.

Next, the time is 24 hours, which is 86,400 seconds. So speed is distance divided by time. 40,074 km divided by 86,400 seconds. Let me do that division: 40,074 / 86,400 ≈ 0.463 km/s. So the order of magnitude is 10^0, since 0.463 is between 0.1 and 1. But wait, maybe I should consider uncertainties. The Earth's radius might not be exactly 6,371 km. Maybe it's a bit more or less. Also, the exact value of π is 3.1416, but using 3.14 would be close enough. Let me check: 2 * 3.14 * 6371 ≈ 40,030 km. So the circumference is about 40,000 km. Dividing by 86,400 seconds gives approximately 0.463 km/s. So the answer is roughly 0.46 km/s, which is 4.6 x 10^-1. The order of magnitude is 10^(-1) because 0.46 is 4.6 * 10^-1. So the exponent is -1.

Now, considering the probabilistic program. The main factors are the Earth's circumference and the time. The circumference could vary slightly. Let's model the circumference as a Gaussian with mean 40,000 km and maybe a standard deviation of 1,000 km. The time is 24 hours, which is precise, but maybe there's some uncertainty in the exact time Santa has. However, the problem says "24-hour period," so maybe it's exact. Alternatively, maybe Santa has a little extra time, but that's probably negligible. So the main uncertainty is in the circumference. Then, the speed would be circumference divided by 86,400 seconds. So the program would be a Gaussian for the circumference, then divide by 86,400. But since the question asks for the order of magnitude, maybe the program should reflect that the answer is around 0.5 km/s, which is 5 x 10^-1. So the exponent is -1. Therefore, the program should have a Gaussian with mean -1 and maybe a small sigma. But wait, the problem says to express the answer as a real number, not the exponent. Wait, no. The user wants the base-10 exponent. Wait, the user says, "Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million)." So the answer is the exponent, like 6 for 1e6. So in this case, the answer is approximately 0.46 km/s, which is 4.6e-1, so the exponent is -1. So the final answer is -1. The probabilistic program should model that. So maybe the program is a Gaussian with mean -1 and some small sigma. But how to model the uncertainty? The circumference could be slightly different, leading to a different speed. Let's say the circumference is 40,000 km with some variation. Let me write the program. The circumference is 2 * π * radius. Let's model the radius as a Gaussian with mean 6371 km and some standard deviation. Then, the circumference is 2 * π * radius. Then, divide by 86400 seconds. But since the problem is about order of magnitude, maybe the main uncertainty is in the radius. Alternatively, maybe the circumference is 40,000 km, and the time is 24 hours. So the speed is 40,000 / 86400 ≈ 0.463 km/s. So the exponent is -1. So the program could be a Gaussian with mean -1 and a small sigma, like 0.1. But the user wants the program to reflect uncertainty. So perhaps the program is (gaussian -1 0.1). But I need to write the program in Pluck. Let me check the allowed constructs. The program must evaluate to a real number, which is the answer in km/s. So the program would be something like (gaussian -1 0.1). Wait, but the actual value is around 0.46, which is 4.6e-1, so the exponent is -1. So the answer is -1. Therefore, the program is (gaussian -1 0.1). But maybe there's more uncertainty. For example, if the Earth's circumference is 40,000 km, but maybe it's 40,000 ± 1,000 km. Then, the speed would be 40,000 / 86400 ≈ 0.463, and 40,000 + 1,000 is 41,000, which would be 41,000 / 86400 ≈ 0.475, so the exponent is still -1. So the standard deviation could be 0.1. So the program is (gaussian -1 0.1). That would model the uncertainty around the exponent. So the final answer is -1.

③ Gold label: y = -1.0000 (log10)

④ Model completion (answer after reasoning):

<program>
(gaussian -1 0.1)
</program>

Parsed → induced: 1.00·N(-1.00, 0.10); log-score 1.384

✗ Rejected — program/mixture emitted but rejected

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: It's difficult to get seafood to the most landlocked state in the nation: Nebraska. How much gasoline would it take to transport one ton of salmon from Los Angeles to Omaha, Nebraska? Provide this answer in terms of dollars (cost of the gasoline it would take), using the average price of California gasoline.

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, let's tackle this Fermi problem. The question is about estimating the cost of gasoline to transport one ton of salmon from LA to Omaha, Nebraska. First, I need to figure out the distance. Nebraska is landlocked, so Omaha is probably around 1000 miles from LA? Wait, maybe 1200 miles? Let me check... Well, I don't have exact data, so I'll estimate. Let's say about 1200 miles.

Next, fuel efficiency. A truck's fuel efficiency is key. Let's assume a typical truck gets about 6 miles per gallon. So, 1200 miles divided by 6 mpg gives 200 gallons. But wait, that's a rough estimate. Maybe the truck is more efficient? Maybe 8 mpg? Then 150 gallons. Hmm, but I should consider that maybe the truck isn't full, so maybe 100 gallons? Wait, but I need to be careful here. Let's say 100 gallons as a rough estimate.

Now, cost of gasoline. Average price in California. Let's say around $3.50 per gallon. So 100 gallons would be $350. But this is a lot. Wait, maybe I'm missing something. Let me think again.

Wait, the distance from LA to Omaha is actually less. Maybe around 1000 miles? Let me think. LA to Omaha is roughly 1000 miles. Let me check. If it's 1000 miles, then 1000 / 6 = 166.67 gallons. At $3.50, that's about $583. But maybe the truck is more efficient. Let's say 8 mpg, so 125 gallons. Then $437.50. But I'm not sure. Also, maybe the truck isn't driven at full capacity, so maybe the fuel used is less? Or maybe the distance is shorter. Maybe 800 miles? Then 800 / 6 = 133 gallons. At $3.50, that's $466. 

Alternatively, maybe the truck is 10 mpg. Then 800 / 10 = 80 gallons. $280. But I need to be cautious. Let's go with 100 gallons as a base. So the cost is around $350. 

But the problem says "order of magnitude," so maybe I should think in terms of powers of ten. If the answer is around $350, that's 10^2 dollars. But maybe the actual value is higher. Maybe the distance is longer, or the fuel efficiency is lower. 

Alternatively, let's think of the order of magnitude. If the answer is around $350, then the order of magnitude is 10^2. But maybe I'm underestimating. For example, if the distance is 1200 miles, and the truck is 6 mpg, then 200 gallons. At $3.50, that's $700. That's 10^2 as well. So maybe the answer is in the hundreds of dollars, which is 10^2. 

But I need to model this with uncertainty. So, the variables are distance, fuel efficiency, and gas price. Each of these has some uncertainty. Let me model them as Gaussian variables. 

Distance: Let's say 1000 miles with some variation. Maybe 1000 ± 200 miles. So a Gaussian around 1000 with sigma 200. 

Fuel efficiency: Let's say 6-10 mpg. Maybe average 7 mpg. So a Gaussian around 7 with sigma 1.5. 

Gas price: Let's say $3.50 with some variation. Maybe $3.00 to $4.00. So Gaussian around 3.5 with sigma 0.5. 

Then, the calculation is (distance / fuel efficiency) * gas price. 

So the total cost would be (distance / fuel_efficiency) * gas_price. 

Let me compute that. 

But since these are variables, I need to model them as Gaussian variables. 

But how to combine them? Let's see. The distance is D, fuel efficiency is F, gas price is P. 

Total cost is (D / F) * P. 

So, if D is Gaussian with mean 1000, sigma 200, F is Gaussian with mean 7, sigma 1.5, P is Gaussian with mean 3.5, sigma 0.5. 

But since these are independent variables, the product would be a more complex distribution. However, for order of magnitude, maybe the product is around (1000 / 7) * 3.5 = (142.86) * 3.5 ≈ 500. So that's 500, which is 10^2. 

But considering variations, if D is 1000, F is 7, P is 3.5, then the cost is 500. If D is 1200, F is 6, P is 3.5, then (1200/6)*3.5 = 200 * 3.5 = 700. Which is 7*10^2. If D is 800, F is 10, P is 3.5, then (800/10)*3.5= 80*3.5=280, which is 2.8*10^2. 

So the order of magnitude is around 10^2. However, if variables vary more, maybe it's 10^2 or 10^3. But given that the answer is in the hundreds, 10^2 is the order of magnitude. 

But the problem asks for the order of magnitude as an integer exponent. So the answer is 2. 

But let me check again. If the answer is $350, then it's 10^2. If it's $700, it's still 10^2. If it's $1000, that's 10^3. But based on the calculation, I think the answer is in the hundreds, so 10^2. 

So the final answer is 2. 

Now, the probabilistic program. The program needs to model the uncertainty in distance, fuel efficiency, and gas price. Let's model each as a Gaussian. Then compute the cost. 

So, the program would be:

(let ((distance (gaussian 1000.0 200.0))
      (fuel-efficiency (gaussian 7.0 1.5))
      (gas-price (gaussian 3.5 0.5)))
  (* (/ distance fuel-efficiency) gas-price))

But I need to make sure that the variables are correctly modeled. However, since the problem allows for flip+if branches for scenarios, maybe I should consider different scenarios. For example, maybe the distance is longer or shorter. But since the problem is about order of magnitude, maybe the Gaussian is sufficient. 

Alternatively, use flip for different scenarios. But the question says to use flip+if when qualitatively different scenarios are plausible. 

But perhaps the main variables are distance, fuel efficiency, and gas price. So the program would be:

(let ((distance (gaussian 1000.0 200.0))
      (fuel-efficiency (gaussian 7.0 1.5))
      (gas-price (gaussian 3.5 0.5)))
  (* (/ distance fuel-efficiency) gas-price))

But the problem says to use flip+if when different scenarios are plausible. For example, maybe the truck is more efficient or less. But the Gaussian already accounts for that. 

Alternatively, maybe the distance is 1000 miles (mean) with some variation. 

So the final answer is 2. The order of magnitude is 10^2.

③ Gold label: y = 2.0000 (log10)

④ Model completion (answer after reasoning):

<program>
(let ((distance (gaussian 1000.0 200.0))
      (fuel-efficiency (gaussian 7.0 1.5))
      (gas-price (gaussian 3.5 0.5)))
  (*. 1.0 (/ distance fuel-efficiency) gas-price))
</program>

Outcome: answer emitted but the scorer rejected it — PanicException: unknown token: '/' with env ["gas-price", "fuel-efficiency", "distance"] → counted as a parse failure (reward floor).

✗ Overrun — reasoning overran, no answer

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: How many cups of tea is consumed around the world in a year?

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, let's tackle this Fermi question. The user wants an order of magnitude estimate for the number of cups of tea consumed worldwide in a year. First, I need to break it down. 

I know that the global population is around 8 billion. Let's say about 8e9 people. Now, how many cups of tea does an average person drink per year? Maybe 2-3 cups a day? Let's go with 2 cups a day. That would be 2 * 365 = 730 cups per person per year. Multiply that by 8e9 people: 8e9 * 730 = 5.84e12. But wait, that's a rough estimate. Maybe the average is lower. Let me adjust. Maybe 1 cup a day? That would be 365 per person, so 8e9 * 365 = 2.92e12. But I'm not sure if that's accurate. 

Alternatively, I remember that in some countries, tea consumption is higher, like in India or China. Maybe the average is around 1.5 cups per day. So 1.5 * 365 = 547.5. Then 8e9 * 547.5 = 4.38e12. But I'm not certain. Also, maybe some people don't drink tea at all. Let's say 50% of the population drinks tea. That would halve the number. So 4.38e12 / 2 = 2.19e12. 

But how to model this uncertainty? The answer is probably in the range of 1e12 to 1e13. So the order of magnitude is 10^12. But I need to write a probabilistic program. Let's structure it. 

Start with population: 8e9. Then, average cups per person per day. Let's model that as a Gaussian, maybe 2.0 with some sigma. Then multiply by 365. Then, account for possible variations in population, like 8e9 ± 1e9. Also, maybe some scenarios where the average is lower, like 1.5 or 2.5. So use a flip to choose between different scenarios. 

Wait, the user wants the base-10 exponent. So if the answer is around 1e12, the exponent is 12. But let's check. If the answer is 5e12, that's 10^12.5, but the order of magnitude is 10^12. So the answer would be 12. 

But the program needs to be a generative model. So I'll model the population as a Gaussian around 8e9 with some variance. Then, the average cups per person per day as a Gaussian around 2.0. Multiply by 365. Then, maybe a flip for scenarios where people drink less. 

Alternatively, use a flip to choose between different average consumption rates. For example, a 50% chance of 1 cup per day and 50% of 2 cups. Then compute the total. 

But I need to make sure the program reflects my uncertainty. So maybe:

Population is 8e9 (but with some variance). Average per person per day is a Gaussian around 1.5 with sigma 0.5. Multiply by 365. Then, maybe a flip for 10% chance of a higher consumption. 

Alternatively, the program could be:

Let population be a Gaussian with mean 8e9 and sigma 1e9. Let average_cups be a Gaussian with mean 2.0 and sigma 0.5. Then total is population * average_cups * 365. But since the user wants the exponent, maybe the program should output a value that's in the 1e12 range. 

But the exact answer might vary. Let's think of the most probable value. If 8e9 people, 2 cups per day, that's 8e9 * 2 * 365 = 5.84e12. So the exponent is 12. But if the answer is 5e12, which is 10^12.7, but the order of magnitude is 10^12. So the answer is 12. 

So the final answer is 12. The program should model that. Maybe using a Gaussian with mean 12 and some sigma. But the user wants the program to represent the uncertainty. So perhaps:

(let ((population (gaussian 8.0 1.0))) ; 8e9
      (average (gaussian 2.0 0.5))
      (total (* population average 365.0)))
But that would be in terms of 1e9. Wait, population is 8e9, so 8.0e9. But the program should have numbers with decimal points. So 8.0e9 is 8000000000.0. But the program would have to compute that. 

Alternatively, the program could be:

(let ((population (gaussian 8.0 1.0)) ; 8e9
      (daily (gaussian 2.0 0.5)))
  (*. 365.0 (* population daily)))

But the problem is that the program should evaluate to a number, but with the log-density, the user wants the program to reflect the uncertainty. So maybe the correct approach is to model the population as a Gaussian around 8e9, average cups per day as 2, and then multiply by 365. The result would be around 5.8e12, which is 10^12. So the exponent is 12. 

But the program needs to be written in Pluck. So the final answer is 12. Therefore, the program would be a Gaussian with mean 12 and some sigma. But perhaps the correct way is to model the total as a Gaussian with mean 12 and sigma 1. 

Wait, but the user is asking for the base-10 exponent. So the program should output a number that is the exponent. For example, if the true value is 5e12, the exponent is 12. 

But the program is a generative model of the answer. So the program should compute the answer in the units of cups, then take the log10 of that. But the problem says to express the answer as the base-10 exponent. 

Wait, the user says: "the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors."

So the program should compute the number of cups, then the exponent is the log10 of that number. But how to model that? 

Alternatively, the program's output is the number of cups, and the user wants the exponent. But the program is supposed to model the uncertainty in the number of cups. So the program would compute the number of cups, and then the exponent would be log10 of that. But how to represent that in the program. 

But the problem says to output the base-10 exponent as an integer. So the program should generate a number, and then the exponent is the log10 of that number. But that's not directly possible. 

Alternatively, the program is supposed to generate a number, and the user is to find the exponent. But the program is a model of the answer. So if the answer is around 5e12, the exponent is 12. 

So the final answer is 12. Therefore, the program would be a Gaussian with mean 12 and some sigma. 

But how to model the uncertainty? Maybe the program is:

(let ((population (gaussian 8.0 1.0)) ; 8e9
      (daily (gaussian 2.0 0.5)))
  (*. 365.0 (* population daily)))

But this would compute the total number of cups, which is around 5.8e12. So the exponent is 12. But the program should model the uncertainty. So maybe the program is a Gaussian with mean 12 and sigma 1. 

But the user wants the program to be in Pluck. So the correct approach is to model the total as a Gaussian with mean 12 and some standard deviation. 

But perhaps the answer is 12. So the program would be:

(gaussian 12.0 1.0)

But that's a bit simplistic. Alternatively, considering different scenarios. For example, a flip for different average consumption rates. 

So the program could be:

(let ((population (gaussian 8.0 1.0)) ; 8e9
      (average (flip 0.5) ; 50% chance of 1.5 or 2.0
        (if (flip 0.5) (gaussian 1.5 0.5) (gaussian 2.0 0.5))))
  (*. 365.0 (* population average)))

But that's more complex. But the user wants the program to reflect uncertainty. 

Alternatively, the program could be:

(let ((population (gaussian 8.0 1.0)) ; 8e9
      (average (gaussian 2.0 0.5)))
  (*.

③ Gold label: y = 12.0000 (log10)

④ Model completion (answer after reasoning):

[no </think> — reasoning overran the token budget; no answer emitted]

Outcome: no answer block — <think> consumed the token budget → parse failure (reward floor).

survival · trained pluck (ckpt 300) — parse rate 1.00, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating a time-to-event outcome in the gbsg survival cohort. The feature columns are anonymized; reason from the marginal distribution and any plausible scale you can infer from the cohort name.

Patient features:
  x0: 1
  x1: 1
  x2: 0
  x3: 51
  x4: 1
  x5: 167
  x6: 109

Provide your estimate of the natural log of the time to event in the same time unit as the cohort (days for SUPPORT, months for METABRIC and GBSG).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the time to event in the GBSG cohort. The features given are anonymized, so I can't use them directly. The cohort is GBSG, which is known for survival data, typically in months. The question asks for the natural log of the time to event in the same units as the cohort, so probably months.

The user wants a probabilistic program in Pluck. I should model the log time as a Gaussian distribution. Since I don't have specific data, I'll make an educated guess. GBSG data might have a mean log time around 5-6 months, but I need to check if that makes sense. Wait, if the event time is in months, taking the log would make it more manageable. Let's assume a mean of 5.0 and a standard deviation of 1.5, which is a common choice for such models. I'll use a Gaussian distribution with those parameters. The program should reflect uncertainty, so maybe a simple Gaussian without branching unless there's a reason to think different scenarios. Since the features are anonymized, I can't account for specific variables, so the model is straightforward. The final answer is the Gaussian distribution with mean 5.0 and sigma 1.5.

③ Gold label: y = 2.3589 (log time) [right-censored]

④ Model completion (answer after reasoning):

<program>
(gaussian 5.0 1.5)
</program>

Parsed → induced: 1.00·N(5.00, 1.50); log-score -0.040

fermieval · trained pluck (ckpt 300) — parse rate 1.00, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
Estimate the order of magnitude of the answer to the following Fermi-style question.

Question: When written in Fahrenheit, what is the sum of the temperatures in Cambridge, Massachusetts, at 8 a.m. every day of the last year?

Provide your estimate of the base-10 exponent (an integer between -100 and 100; e.g. 6 means around one million).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, let's tackle this Fermi question. The user wants the order of magnitude of the sum of temperatures in Cambridge, MA, in Fahrenheit at 8 a.m. every day of the last year. First, I need to estimate the average daily temperature. Cambridge's typical daily high is around 60-70°F, but 8 a.m. is probably a bit cooler, maybe 50-60°F. Let's say 55°F on average. Over 365 days, that's 55 * 365 ≈ 20,000°F. The order of magnitude here is 10^4 (since 20,000 is 2x10^4). But considering possible variation, like some days being colder or warmer, maybe the sum could be around 10^4. So the exponent is 4. The probabilistic program should reflect this with a Gaussian around 4 with some sigma. Maybe 4.0 ± 1.0. So the program is a Gaussian with mean 4 and sigma 1.

③ Gold label: y = 4.0000 (log10)

④ Model completion (answer after reasoning):

<program>
(gaussian 4.0 1.0)
</program>

Parsed → induced: 1.00·N(4.00, 1.00); log-score -0.919

ames_housing · trained pluck (ckpt 300) — parse rate 0.99, n=200

✓ Good — best-scoring parsed rollout

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating the sale price of a residential property in Ames, Iowa (sales 2006-2010).

Property features:
- Neighborhood: Mitchell
- MS_Zoning: Residential_Medium_Density
- House_Style: One_Story
- Year_Built: 1999
- Year_Remod_Add: 1999
- Overall_Qual: Above_Average
- Overall_Cond: Average
- Gr_Liv_Area: 1224
- Total_Bsmt_SF: 1224
- Garage_Type: Attchd
- Garage_Cars: 2
- Full_Bath: 2
- Half_Bath: 0
- Bedroom_AbvGr: 2
- Pool_QC: No_Pool
- Fireplaces: 0
- Lot_Area: 4500
- Central_Air: Y

Provide your estimate of the natural log of the sale price in USD (typical values 10.5 to 13.5).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the sale price for a residential property in Ames, Iowa from 2006-2010. The property is in the Mitchell neighborhood, which I recall is a mid-range area. The MS_Zoning is Residential_Medium_Density, so that's a moderate zoning. The house is a one-story, built in 1999, same year it was remodeled. Overall quality is above average, but condition is average. The square footage is 1224, both above ground and basement. Garage is attached, two cars, two baths, two bedrooms. No pool, no fireplaces. Lot area is 4500 sq ft. Central air is yes.

First, I should consider typical sale prices in that area. The user mentioned typical values are between 10.5 and 13.5 for the natural log. So the average might be around 12.0. But factors like the neighborhood, condition, and features might affect it. Since the house is in a medium-density zone, maybe a bit higher. The overall quality is above average, which could add a bit. The garage is attached and two cars, which is good. The lot size is decent, but no pool or fireplaces might be a downside. Central air is a plus. The year built is 1999, which is not too old. The condition is average, so maybe that's a slight drag. The square footage is 1224, which is a bit on the lower side for a house, but maybe in a good neighborhood.

Considering all these, I think the log price is around 12.2. But there's some uncertainty. Maybe a range of 11.8 to 12.6. So I can model it as a Gaussian with mean 12.2 and standard deviation 0.3. But I should check if there are any rare scenarios. For example, if the neighborhood is less desirable, maybe a lower value. But Mitchell is a typical neighborhood, so that's probably not a factor. The condition being average might slightly lower it, but the quality is above average. So the main factors are the average condition and the above-average quality. Maybe the standard deviation is 0.3. So the program would be a Gaussian with mean 12.2 and sigma 0.3. But I need to express it as a probabilistic program. So the final answer is a Gaussian with those parameters.

③ Gold label: y = 12.0137 (log USD)

④ Model completion (answer after reasoning):

<program>
(gaussian 12.2 0.3)
</program>

Parsed → induced: 1.00·N(12.20, 0.30); log-score 0.092

✗ Rejected — program/mixture emitted but rejected

① Prompt (system + question + answer-format instruction):

[SYSTEM]
You are a careful estimator. Briefly reason about the question, then produce your final answer in the exact format requested. Keep reasoning concise (a few sentences max).

[USER]
You are estimating the sale price of a residential property in Ames, Iowa (sales 2006-2010).

Property features:
- Neighborhood: College_Creek
- MS_Zoning: Residential_Low_Density
- House_Style: One_Story
- Year_Built: 2006
- Year_Remod_Add: 2007
- Overall_Qual: Very_Good
- Overall_Cond: Average
- Gr_Liv_Area: 2140
- Total_Bsmt_SF: 2140
- Garage_Type: Attchd
- Garage_Cars: 3
- Full_Bath: 2
- Half_Bath: 1
- Bedroom_AbvGr: 3
- Pool_QC: No_Pool
- Fireplaces: 0
- Lot_Area: 10481
- Central_Air: Y

Provide your estimate of the natural log of the sale price in USD (typical values 10.5 to 13.5).

Express your belief as a small probabilistic program in Pluck (a Scheme-like language). The program is a generative model of the answer; your score is the log-density of the true value under the distribution your program induces, so make the program reflect your honest uncertainty.

Allowed constructs (write all numbers with a decimal point, e.g. 5.0):
  (gaussian MU SIGMA)       continuous quantity: mean MU, std dev SIGMA
  (flip P)                  scenario coin-flip with probability P
  (if C A B)                branch on a flip
  (let ((x EXPR) ...) BODY) name intermediate quantities
  (+. a b), (-. a b)        add/subtract continuous quantities
  (*. C a)                  scale by a numeric constant C only; never multiply two gaussians

The final expression must evaluate to one real number: the answer in the units requested above. Use flip+if branches when qualitatively different scenarios are plausible; sum gaussians to combine independent uncertain factors.

Output exactly one program between tags, like:
<program>
(let ((typical (gaussian 5.0 1.0))
      (rare (flip 0.2)))
  (if rare (gaussian 9.0 2.0) typical))
</program>

② Model reasoning (<think>):

Okay, I need to estimate the natural log of the sale price for a residential property in Ames, Iowa from 2006-2010. The features given include a neighborhood (College_Creek), which is a residential area with low density. The house is a one-story, built in 2006, remodeled in 2007. The overall quality is very good, but the condition is average. The living area is 2140 sq ft, with 2140 sq ft basement. The garage is attached with 3 cars, 2 full baths, 1 half bath, 3 bedrooms, no pool, no fireplace, and a lot size of 10481 sq ft. Central air is present.

First, I should consider typical sale prices in that area during that time. The user mentioned typical values are between 10.5 and 13.5 for the natural log. Since the house is in College_Creek, which is a residential neighborhood, and the house is in good condition but not top-tier, maybe the price is mid-range. The size is decent, 2140 sq ft, but with a basement, which adds value. The garage is attached with 3 cars, which is good. No pool or fireplace might slightly lower the price, but central air is a plus.

I think the natural log of the sale price would be around 12.0. But there's uncertainty. Maybe a 1.0 standard deviation. However, since the overall condition is average, maybe it's a bit lower, like 11.5. But the house is very good in quality, so maybe 12.0. So a Gaussian around 12.0 with a standard deviation of 0.5. But I should also consider if there's a rare scenario where the price is higher or lower. Maybe a 0.2 chance of being higher, like 12.5, or lower, like 11.5. So the program could be a Gaussian with mean 12.0 and sigma 0.5, with a small chance of flipping to another value. But since the user says to use flip+if when scenarios are plausible, maybe a 10% chance of being 11.5 or 12.5. So the final program would be a Gaussian around 12.0 with sigma 0.5, and a 0.1 chance of adjusting it slightly.

③ Gold label: y = 12.7939 (log USD)

④ Model completion (answer after reasoning):

<program>
(let ((base (gaussian 12.0 0.5))
      (adjust (flip 0.1)))
  (if adjust (+. base (-. 0.5)) base))
</program>

Outcome: answer emitted but the scorer rejected it — PanicException: assertion `left == right` failed: FSub expected 2 arguments, got 1 left: 1 right: 2 → counted as a parse failure (reward floor).

Model-size scaling — does a bigger model use pluck's structure?

The central finding (pluck never realizes structure) could be a capacity limit. To test it, the same survival pluck GRPO was run at 4B, 8B, and 14B (the largest that fits this 96 GB GPU under vLLM colocate). Held-out 200 cells, best checkpoint each:

model	log-score mean ↑	median ↑	CRPS ↓	cov 90%	structure used
Qwen3-4B pluck	−2.099	−1.793	1.251	0.81	branch≡0, ncomp ~1.8
Qwen3-8B pluck	−2.135	−1.823	1.306	0.77	branch≡0, ncomp ~1.8
Qwen3-14B pluck	−1.976	−1.767	1.060	0.86	branch≡0, ncomp ~1.0
Qwen3-4B MoG (reference)	−1.994	−1.794	—	0.83	—

Not capacity — reward. The no-structure result holds at every scale. branch (point-mass / label mass) is identically 0 at 4B, 8B, and 14B — and 14B uses the least structure of all, collapsing to a single gaussian (ncomp ~1.0) rather than the smaller models' mixtures. Scaling 3.5× (4B→14B) buys only ~0.12 nat on the mean (a 14B pluck ≈ a 4B MoG), with the gain in calibration/CRPS, not structure. So pluck's unrealized expressivity is reward-bound, not capacity-bound: a bigger model does not discover point masses or factor composition under the log-density signal. Realizing PPL structure needs a reward or exploration signal that pays for it — the cure-exemplar probe is the cheap exploration lever; a structure-aware reward is the principled one. (32B would not fit colocate on one GPU; sequence length was never the constraint — prompts are 350–510 tokens.)

Engineering notes

Pluck runs sandboxed: a worker process survives Rust panics; a deliberate 2²²-path timeout bomb is killed and respawned in <5s; steady-state ~0.6 ms/program.
Bindings built into the repo-root venv on Euler (rustup user-local + maturin develop --release) — generation, scoring, training, and eval all run from one node since today; the Mac round-trip is retired.
The whole session is one idempotent script: bash training/gpu_session.sh 300 (skip-if-done pluck generation → in-place scoring → three sequential resumable trainings).

Open questions

Pluck GRPO go/no-go: trained MoG-XML now beats the untrained pluck baseline even on survival (−1.99 vs −6.66), so the fair fight is trained-vs-trained — does pluck training close the gap, and is the format curriculum (few-shot scaffolding for the 0.47 fermieval parse rate) worth building first? Should the language accept scientific notation?
Fresh post-cutoff Metaculus pull (needs API token) to replace the leaky split.
what_are_the_odds label regeneration → would add a 5th headline dataset.

Autonomous pluck session (2026-06-18, in progress)

Updated 2026-06-18, Euler node eu-g7-007 (RTX PRO 6000 Blackwell, 24 h alloc). Autonomous overnight session: Phase 4 — pluck GRPO ("pluck go"), plus two report upgrades (matched-subset reporting, full-detail rollout traces). This tab and the Regression-PPL tab are kept live as results land.

Now running

Experiment	Status	Detail
E1 — pluck v1 GRPO (survival, fermieval, ames, 300 steps, LoRA)	all 3 done & evaluated	parse → 1.00 everywhere; trained pluck ties-or-beats trained MoG on all three (table below).
E3 — cure-exemplar probe (survival, `pluck_cure` prompt)	done	best survival model overall (−1.853, beats MoG −1.99 & v1 pluck −2.10) — but the gain is on uncensored rows and point masses are mis-placed; structure reachable, not correctly used.
E4 — variance control (v1 pluck survival re-run, same config)	done	v1 reruns at −2.10…−2.22 (variance ≈0.1 nat); cure −1.85 beats it beyond variance → the cure gain is real (in the mean/tail; median unchanged).

E5 — cure variance check (2nd pluck_cure survival run) done — reproduced both cure runs (−1.85, −1.92) beat both v1 runs (−2.10, −2.22); gap real (~0.27 nat). E6 — bigger model: 8B (Qwen3-8B pluck, survival) done capacity test answer: NO structure — branch≡0.000 through all 300 steps, like 4B (ncomp~1.8 gaussian mixtures). Best raw-ls −2.06 ≈ 4B. Held-out eval pending GPU. E6 — bigger model: 14B (Qwen3-14B pluck, survival) done & evaluated fits at util 0.45 (~78 GB, no OOM). branch≡0, ncomp collapsed to ~1.0; held-out −1.976 (best pluck, ≈ a 4B MoG).

Model-size scaling on survival (held-out, best ckpt)

model	ls-mean ↑	median ↑	CRPS ↓	cov90	structure
4B pluck	−2.099	−1.793	1.251	0.81	branch≡0, ncomp~1.8
8B pluck	−2.135	−1.823	1.306	0.77	branch≡0, ncomp~1.8
14B pluck	−1.976	−1.767	1.060	0.86	branch≡0, ncomp~1.0

Capacity hypothesis rejected across 4B/8B/14B: a bigger model does not unlock pluck's structure (branch≡0 everywhere; 14B uses the least). Scaling 3.5× buys ~0.12 nat (14B pluck ≈ 4B MoG) — diminishing calibration gains, no structural emergence. The limit is the reward, not model size. GPU now free.

Phase 4 campaign conclusion (E1–E4, E5 confirming). Trained pluck ties-or-beats trained MoG everywhere as an exact-inference engine; the cure exemplar gives a real extra gain but the model never deploys structure correctly — realizing PPL structure needs a reward that pays for it. Full write-up in the Regression-PPL tab and 5-regression-ppl/phase4_log.md.

Headline — trained pluck vs trained MoG (matched, full held-out 200)

dataset · series	ls-mean ↑	median ↑	abs-err ↓	CRPS ↓	cov 90%
survival · trained MoG	−1.99	−1.79	2.61	1.71	0.83
survival · trained pluck	−2.10	−1.79	1.79	1.25	0.81
fermieval · trained MoG	−7.11	−1.62	4.20	3.52	0.78
fermieval · trained pluck	−4.93	−1.61	3.26	2.80	0.80
ames · trained MoG	−0.567	−0.440	0.314	0.229	0.955
ames · trained pluck	−0.555	−0.421	0.304	0.220	0.950

Survival: trade-off (tied median log-score, MoG +0.1 nat mean, pluck −31% abs-err / −27% CRPS). fermieval: pluck wins outright on the full holdout (its single wide gaussian is adaptive for heavy tails). In both, pluck never uses point-mass / factor structure — it collapses to the gaussian subset MoG covers, so the program format helps as an inference engine, not via richer structure. Full tables + the censored-row structure probe in the Regression-PPL tab.

Done this session

Item	Status	Outcome
Matched-subset reporting (project-wide standard)	done	`matched_report.py`; Phase 3 table rebuilt on matched subsets + parse rates
Full-detail expandable rollout traces	done	each example: prompt → reasoning → gold → completion; good + both failure modes
Pluck GRPO trainer built + self-tested	done	`reward_pluck.py` (caches inference, ~1 ms/rollout), `train_pluck.py`, `pluck_session.sh`, `eval_pluck.sh`
E2 — pluck_v2 improved-prompt baseline	negative — dropped	v2 prompt HURT zero-shot (survival parse 0.70→0.33; log-space framing confused the model). Cheap baseline killed the v2 GRPO branch, saving ~1.8 h GPU.

Findings so far

GRPO lifts pluck parse rate on its own (survival 0.70 base → 1.00 by step ~26), just like MoG — no format curriculum needed.
Trained pluck ≈ a MoG expressed as a program. Reading rollouts (not just the metric): 28/32 programs at step 50 use flip/if, but they branch between two gaussians → a gaussian mixture (ncomp≈1.8). The model does not use point-mass "cure" leaves (the one structure a MoG can't express). So the open question sharpens to: does a program-mixture beat an XML-mixture when both are trained?
Prompt complexity ≠ better (E2): the base model needs fewer rules, not more; GRPO, not prompting, is what fixes the variadic-+. and division failures.

Queued (to keep the GPU busy)

Merge + held-out 200-cell eval for each pluck dataset as it finishes (eval_pluck.sh).
Trained-vs-trained MoG vs pluck on matched subsets — the headline Phase 4 result.
Structure probe: on censored survival rows a point-mass cure leaf is exactly right and a gaussian MoG cannot express it — does trained pluck use it, or leave the free lunch? (Split survival eval by censored vs not.)
If time: one 600-step pluck run on fermieval (the multiplicative / log-space-sum case where pluck's (+.) composition could most plausibly beat MoG).

openestimate dropped as a 4th training dataset — only 256 examples, 56 left after the 200-cell holdout. Campaign log: 5-regression-ppl/phase4_log.md.

Method	Output	Induced density
scalar (baseline)	`<answer>V</answer>`	\( \mathcal N(V, \hat\sigma) \), \(\hat\sigma\) fit on train residuals — richer formats must beat this
beta	`<beta><alpha>A</alpha><beta>B</beta></beta>`	Beta(A,B) on range-normalized target (Jacobian-corrected)
calibrated_ci	90% interval `[L,U]`	\( \mathcal N\!\bigl(\tfrac{L+U}{2}, \tfrac{U-L}{2\cdot1.645}\bigr) \)
mog_xml ← Phase 3 target	`<mixture><component><mu><sigma><w>…`	\( \sum_k w_k \,\mathcal N(y;\mu_k,\sigma_k) \), K ≤ 4
pluck ← Phase 4	Scheme-like program (below)	exact-inference marginal: Gaussian + point-mass components