{"id":"3b33d97c-8ea2-4cee-b283-576a6b9cc3d6","arxiv_id":"2506.21964","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"LLMs suggested directionally correct but poorly calibrated Bayesian priors, with Claude's weak priors ranking best on KL divergence from the data.","lead":"Researchers tested three large language models as a way to automatically suggest Bayesian prior distributions, the starting guesses used in statistical modeling, for two real datasets. The models usually knew the direction of each effect, but often set their guesses too narrowly or too widely, so calibrating uncertainty remains the main problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Quantitative LLM ranking rests on a single noisy KL-to-MLE benchmark; the ordering is not yet load-bearing.","rationale":"The paper's qualitative contribution is real: the authors built a structured elicitation prompt, ran three LLMs on two real datasets, and made a modest, plausible observation that LLMs know signs but misjudge widths. I give credit for that. The point that fails under scrutiny is quantitative: every model comparison is a point estimate derived from one observed dataset, one Gaussian approximation of the MLE, and one LLM output per condition. The reader's verdict already conditions on missing code and repeated runs; my stress-test sharpens that into a specific testable requirement. If the bootstrap stability check shows the ranking is fragile, the abstract's 'Claude and Gemini provided better priors than ChatGPT' and 'significant advantage' would need to be weakened to directional or qualitative statements. If it shows the ranking is stable across resamples and repeated LLM draws, the concern is resolved. Because the suggested fix is additive and the downstream claims are modest, the correct disposition remains CONDITIONAL.","tokens_in":10696,"tokens_out":6866,"duration_ms":87397,"concrete_test":"Bootstrap-resample each dataset 300 times, re-estimate the Gaussian MLE distribution, and recompute all entries in Tables 2 and 5 for the fixed LLM-suggested priors; report the distribution of average-KL ranks. If the top-ranked prior changes in more than 10% of resamples, the ordering in the tables is noise. Then, as a second check, repeat each LLM query 10 times with temperature 0 and 0.7; if the suggested hyperparameters vary across draws, a single response is insufficient to establish model-level differences.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's most concrete claims—Tables 2 and 5, 'Claude and Gemini provided better priors than ChatGPT,' and 'Claude's weakly informative priors were the best'—are all based on D_KL between a Gaussian approximation of the MLE distribution and one set of suggested priors, computed once for each of two datasets. There is no bootstrap, no repeated LLM sampling, and no stated sampling variability for the MLE distribution. In the Cleveland data (n≈303), the asymptotic Gaussian approximation may be poor, and the difference between the top average KL (Claude weak, 1.07) and runner-up (Gemini weak, 1.55) is not assessed against the variability of the benchmark itself. Conceptually, Eq. (3) treats the estimated MLE sampling distribution as a gold standard for the unknown parameter θ; a prior that happened to equal that distribution would receive D_KL=0 and be judged perfect, even though a good knowledge-based prior should reflect pre-data information, not the sampling noise of one study. The qualitative claims—correct signs, frequent overconfidence of moderate priors—are supported by the figures and are likely robust; the quantitative ranking is not. This is a correctness risk for the paper's headline ordering, not for its general direction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes using large language models (LLMs) to elicit informative prior distributions for Bayesian regression models. The authors design a multi-part prompt that asks an LLM to justify its choices, propose both moderately and weakly informative prior sets, and assign confidence scores. They evaluate Claude Opus, Gemini 2.5 Pro, and ChatGPT-4o-mini on two datasets: a Cleveland heart-disease logistic regression and a concrete compressive-strength linear regression. Prior quality is measured by the Kullback-Leibler divergence between a Gaussian approximation of the MLE sampling distribution and each suggested prior. The reported results are that all LLMs identify the correct direction for every association, that moderately informative priors are often overconfident, and that Claude's weakly informative priors achieve the lowest average KL divergence, making Claude and Gemini better than ChatGPT overall. Prediction experiments show only small, statistically insignificant improvements over frequentist baselines.","tokens_in":10928,"tokens_out":3463,"duration_ms":41037,"significance":"If the central claims hold, the paper offers a practical, low-cost route to prior elicitation and identifies a specific failure mode: LLMs are sign-calibrated but width-miscalibrated. The authors deserve credit for explicitly acknowledging the data-bias assumption behind their KL-to-MLE benchmark, for using a structured elicitation prompt with confidence scores, and for including prediction comparisons with a corrected paired t-test. The qualitative findings about direction recall and moderate-prior overconfidence appear well supported by the figures. However, the quantitative ranking of LLMs is currently based on single-run KL values with no uncertainty assessment, and the full prompts and code are not available in the manuscript. These issues make the headline ordering provisional rather than established.","major_comments":[{"comment":"The quantitative rankings rest on KL divergences computed once per LLM and prior set, with no assessment of sampling variability. There is no repeated LLM sampling, no bootstrap or Monte Carlo error for the Gaussian approximation of the MLE distribution, and no sensitivity analysis. For example, in Table 2 the difference between Claude weak (average KL 1.07) and Gemini weak (1.55) is reported as a ranking, but the reader cannot tell whether this gap exceeds the variability of the benchmark itself, and in Table 5 the analogous comparison is even tighter (Gemini moderate 1.09 versus Claude moderate 1.67). To make the central ordering 'Claude and Gemini provided better priors than ChatGPT' and 'Claude's weakly informative priors were the best' load-bearing, the authors should provide uncertainty intervals, repeated LLM draws, and ideally bootstrap resampling of the MLE distribution.","section":"§2.2, Eqs. (2)–(3); Tables 2 and 5"},{"comment":"The KL-to-MLE benchmark assumes that the MLE distribution computed from one observed dataset is a valid reference for prior quality, and the paper explicitly states in Section 4 that the analysis proceeds under the assumption that the data are not severely biased. This assumption is load-bearing: if the Cleveland data carry case-control oversampling (46% prevalence), the disagreement between population-level prior knowledge and the data could reflect data bias rather than poor prior calibration, and all KL-based rankings in Tables 2 and 5 would change. Because the paper itself raises this concern, the authors should add a concrete sensitivity analysis, for instance re-computing rankings with the intercept and sex coefficients excluded, using an alternative reference distribution (e.g., a posterior under a weakly informative prior), or applying prior-predictive checks. As written, the caveat is stated but not operationalized, leaving the main quantitative claim vulnerable.","section":"§2.2 and §4, Eq. (3)"},{"comment":"The exact prompt is a central methodological contribution, but only a skeleton is shown in Listing 1 (with material marked 'not included for brevity') and the code link is deferred to the published version. Without the full prompts, the temperature settings, the number of LLM runs, the date of API access, and the raw outputs, the results cannot be reproduced or assessed for sensitivity to prompt wording. Given that the paper's stated contribution includes 'an extensive prompt,' this material should be included in the supplement or as an appendix, not merely promised for a future version.","section":"§2.1, Listing 1, and 'A link to our code'"},{"comment":"The Kullback-Leibler divergence is used asymmetrically, with the MLE distribution in the numerator and the prior in the denominator. The text interprets this as penalizing priors that are surprised by the data, but the asymmetry also means that very wide priors will receive small KL values regardless of whether they encode useful information. This is not necessarily a flaw, but it directly affects the interpretation of 'unnecessarily vague' priors: ChatGPt's wide weak priors have high KL in Table 2, which seems at odds with the usual behavior of KL for wide distributions. The authors should clarify why the weakly informative ChatGPT priors, described as 'extremely wide,' nevertheless yield some of the largest KL values (e.g., 6.25 for chol in Table 2), since this bears on whether the metric captures overconfidence or something else.","section":"§2.2, Eq. (3) and Figures 2–5"}],"minor_comments":[{"comment":"The model is called 'ChatGPT-4o-mini' in the abstract and Section 3 but 'ChatGPT o4-mini' in Listing 1; reference [10] cites 'Gemini 1.5 Pro' while the experiments use Gemini 2.5 Pro. Please make the model names and citations consistent and include exact version/date information.","section":"Abstract and §3"},{"comment":"The placeholder '(not inclded for the sake of brevity)' appears twice; 'inclded' is a typo and, more importantly, the omitted text is needed for reproducibility. Replace the placeholders with the full prompt or an appendix reference.","section":"Listing 1"},{"comment":"The term 'Kullback-Leibler' is capitalized in the section heading but not in the text; standardize the capitalization. Additionally, the definition in Eq. (3) writes 'DKL(p||pMLE)' while the text sometimes describes the reverse direction; please ensure notation is consistent with the actual direction computed.","section":"§2.2"},{"comment":"The 'Avg Rank' row is reported without explaining how the ranking is computed across variables. Please provide the definition (e.g., rank within each variable across the six prior sets) so the reader can interpret the summary.","section":"Tables 2 and 5"},{"comment":"Reference [9] says '3rd ed edition' (duplicate 'edition') and reference [22] has a stray formatting artifact in the author name. These are minor typographical issues.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's qualitative findings are likely to be robust, but the quantitative ranking is not yet supported because the KL values have no uncertainty quantification and the benchmark's assumptions are only acknowledged, not tested. With a sensitivity analysis and full reproducibility material, the paper could be a useful contribution to the growing LLM-for-statistics literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is an honest, modest paper. LLM-based prior elicitation is not new—Capstick et al. and Gouk & Gao already did it—and the paper's own prediction experiments show no significant gains. What it adds is a clean three-LLM comparison on two real datasets and a robust qualitative finding: the models reliably get the sign of associations right, but their moderately informative priors are often overconfident. That qualitative result is supported by the figures and will likely hold up.\n\nThe reflective prompt—ask the LLM to justify, propose multiple prior sets, and assign confidence weights—is a sensible workflow contribution. The observation that Claude's weak priors kept nonzero means while ChatGPT and Gemini defaulted to zero is a concrete, reproducible-seeming difference. Credit is also due for reporting the null prediction results instead of selling a workflow improvement as a performance win.\n\nNow the soft spots. The stress-test note is fair: the quantitative ranking—'Claude and Gemini better than ChatGPT,' 'Claude weak is best'—is built on KL divergences computed once, with no bootstrap, no repeated LLM sampling, and no uncertainty intervals. On the Cleveland data (n≈303), the gap between average KL 1.07 (Claude weak) and 1.55 (Gemini weak) is plausibly within noise. The KL-to-MLE benchmark is a normative choice; a prior that matches the MLE distribution gets a perfect score, even though a good prior should not be built from one dataset's sampling noise. The authors acknowledge this in Section 4, but they still headline the ranking. Also, the full prompts and code are promised but not actually linked in the arXiv version—a reproducibility gap that a methods paper should not have.\n\nThe prediction section is underpowered and shows no significant improvement; the authors interpret this correctly as expected in large samples, but it means the practical-value claim is speculative.\n\nWho is this for? Bayesian practitioners who want quick, directional priors and anyone working on prior-data conflict diagnostics. It deserves a serious referee: the question is relevant, the qualitative findings are credible, and the quantitative issues are fixable with resampling, uncertainty intervals, and a more careful framing of the benchmark. I'd recommend engaging with it.","headline":"An honest, modest comparison paper whose qualitative direction-recall and overconfidence findings are worth taking seriously, but whose quantitative LLM ranking should not be believed at face value.","tokens_in":11424,"tokens_out":3744,"would_cite":true,"duration_ms":38518,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62B10","62J12"],"pacs":[],"model":"deepseek-v4-flash","headline":"LLMs reliably identify the direction of statistical associations but fail to calibrate the strength of their suggested Bayesian priors, which tend to be overconfident, according to tests on heart-disease and concrete-strength data.","keywords":["large language models","Bayesian statistics","prior elicitation","informative priors","Kullback-Leibler divergence","maximum likelihood estimator","logistic regression","overconfidence"],"falsifier":"Run the same LLM elicitation on simulated datasets whose true regression coefficients are known and that contain no selection bias, then compute how often each suggested prior's central 50% interval covers the true coefficient across many replicates; the paper's overconfidence claim predicts that the moderately informative priors will cover far less than 50% of the time.","tokens_in":10527,"feed_emoji":"📊","tokens_out":8011,"duration_ms":81574,"temperature":0.7,"pith_summary":"Bayesian analysis often settles for vague priors because turning domain knowledge into an actual distribution is difficult, subjective, and time-consuming. This paper tests whether a large language model can do that translation, using a detailed prompt that makes the model justify each hyperparameter, propose both moderately and weakly informative sets, and assign confidence scores. Across a heart-disease logistic regression and a concrete-strength linear regression, the three tested models—Claude, Gemini, and ChatGPT—all identified the correct direction for every association, such as males having higher heart-disease risk. Yet their moderately informative priors, the ones the models themselves trusted most, were often overconfident and sat far from the data, while the weakly informative priors generally agreed better; Claude's weakly informative set was the only one that stayed informative (non-zero mean) and suitably wide. The message is that LLM knowledge of direction is already usable, but calibrating the width of an LLM-suggested prior is the unresolved problem.","feed_headline":"LLMs pick the right sign but the wrong width for Bayesian priors","feed_subtitle":"Across heart-disease and concrete datasets, only Claude's weak priors stayed informative without being overconfident.","key_machinery":"Two mechanisms carry the argument. The first is a structured elicitation prompt: the model is asked to state how it would use domain knowledge, propose at least two prior sets (moderate and weak), justify every normal prior's mean and standard deviation, and rate its own confidence in each set. This is what lets the paper attribute failures to calibration rather than to missing knowledge. The second is the evaluation metric, the Kullback-Leibler (KL) divergence from the Gaussian approximation of the maximum-likelihood estimator's distribution to each proposed prior; because the prior appears in the denominator of the integrand, a prior that is surprised by the data is penalized asymmetrically. Together these mechanisms separate 'does the model know the direction?' from 'does the model know the strength?'.","core_discovery":"The paper's central claim is that LLMs can act as knowledge-based prior elicitors for Bayesian regression only if the width of their suggested distributions is treated with suspicion. In both experiments every model assigned the correct sign to every coefficient, showing that the direction of an association can be reliably extracted from an LLM. However, the 'moderately informative' priors—which the models rated 60-65% confident—frequently placed most of their mass far from the maximum-likelihood estimate, producing high KL divergence; the weakly informative priors generally did better. By the KL-from-MLE benchmark, Claude's weakly informative priors ranked best on both datasets, with average divergences of 1.07 (heart disease) and 2.47 (concrete), while ChatGPT's zero-centered, extremely wide priors ranked worst. The paper concludes that the bottleneck is calibration: the models oscillate between overconfidence in their moderate priors and unnecessarily vague defaults in their weak priors, and that modest predictive gains over a frequentist model were not statistically significant.","pith_inferences":["Going beyond the paper: the same prompt could be run on simulated data with known coefficients, and the coverage of each suggested prior could be measured directly; this would separate general overconfidence from artifacts of the particular datasets used here.","Going beyond the paper: one cheap fix suggested by the results is a post-processing step that keeps the LLM's sign and mean but inflates the standard deviation until the prior predictive overlaps a validation split, borrowing the model's knowledge while forcing data-based scale.","Going beyond the paper: the finding that only Claude avoided the zero-mean default hints that prompt instructions alone do not guarantee calibration-friendly behavior; testing models with identical prompts across repeated runs would show whether this is a stable model difference or a sampling effect.","Going beyond the paper: because the heart-disease data are case-heavy, an unbiased replication would tell whether part of the prior-data gap is the dataset's selection bias rather than the LLM's miscalibration."],"forward_implications":["Practitioners can already use LLM suggestions to fix the sign or direction of coefficients in small-data or regularization settings, where the data alone cannot establish direction.","When an LLM rates a moderately informative prior as its most confident suggestion, that confidence should be discounted; the weakly informative set is the safer choice in the paper's experiments.","In large datasets, LLM priors should not be expected to improve predictive accuracy, because the likelihood dominates the posterior; any benefit is more likely to appear in small samples or out-of-distribution settings.","Prior-data conflict checks based on KL divergence to the MLE distribution can be added to a Bayesian workflow with only the observed data, without needing posterior samples.","Requiring the model to justify a non-zero expected effect prevents the 'unnecessarily vague' zero-mean default that hurt ChatGPT and Gemini's weakly informative priors."],"supporting_citations":[{"why":"The prior LLM-based prior-elicitation work this paper positions itself against; it reported prediction gains the authors could not reproduce.","marker":"[4]"},{"why":"Provides the asymptotic Gaussianity of the MLE that justifies approximating the data distribution by a normal curve.","marker":"[5]"},{"why":"Establishes the prior-data conflict checking tradition that motivates testing whether the data surprise the prior.","marker":"[7]"},{"why":"The standard Bayesian reference that frames how difficult and subjective informative prior elicitation is.","marker":"[9]"},{"why":"Supplies the Cleveland heart disease dataset used in the first experiment.","marker":"[13]"},{"why":"Introduces the use of KL divergence for prior-data conflict, which the paper adapts to compare prior with MLE distribution.","marker":"[19]"},{"why":"Provides the approximate Bayesian inference method used to fit all models in the prediction comparison.","marker":"[22]"},{"why":"Supplies the concrete compressive strength dataset used in the second experiment.","marker":"[25]"}],"fun_headline_variants":["LLMs nail prior direction, miss width","Bayesian priors: LLMs know sign, not scale","Claude wins weak-prior race, but all LLMs flub width","LLM-suggested priors: right sign, wrong spread","AI suggests Bayesian priors, but calibration lags"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the one observed dataset used to compute the MLE distribution is a fair target for a prior, i.e., that the data are not severely biased; if the dataset is biased—for example, the heart-disease sample has a 46% disease prevalence—the KL rankings could be measuring data bias rather than prior quality.","fun_headline_variants_meta":{"raw":{"variants":["LLMs nail prior direction, miss width","Bayesian priors: LLMs know sign, not scale","Claude wins weak-prior race, but all LLMs flub width","LLM-suggested priors: right sign, wrong spread","AI suggests Bayesian priors, but calibration lags"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1592,"prompt_tokens":1006,"completion_tokens":586,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":622,"completion_tokens_details":{"reasoning_tokens":502}},"tokens_in":622,"tokens_out":586,"duration_ms":5470,"temperature":1.0,"reasoning_tokens":502,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:13:47.067652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same LLM elicitation on simulated datasets whose true regression coefficients are known and that contain no selection bias, then compute how often each suggested prior's central 50% interval covers the true coefficient across many replicates; the paper's overconfidence claim predicts that the moderately informative priors will cover far less than 50% of the time.","supporting_citations":[{"cited_title":"Checking for prior-data conflict","cited_arxiv_id":null,"evidence_quote":"Establishes the prior-data conflict checking tradition that motivates testing whether the data surprise the prior."},{"cited_title":"Carlin, Hal S","cited_arxiv_id":null,"evidence_quote":"The standard Bayesian reference that frames how difficult and subjective informative prior elicitation is."},{"cited_title":"Checking for prior-data conflict using prior-to-posterior divergences","cited_arxiv_id":null,"evidence_quote":"Introduces the use of KL divergence for prior-data conflict, which the paper adapts to compare prior with MLE distribution."},{"cited_title":"Approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations","cited_arxiv_id":null,"evidence_quote":"Provides the approximate Bayesian inference method used to fit all models in the prediction comparison."}],"review_version":1}