Extracting Probabilistic Knowledge from Large Language Models for Bayesian Network Parameterization

Aliakbar Nafar; Kristen Brent Venable; Parisa Kordjamshidi; Zijun Cui

REVIEW 3 major objections 5 minor 2 cited by

Extracting Probabilistic Knowledge from Large Language Models for Bayesian Network Parameterization

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read LLM probability guesses can serve as expert priors for Bayesian networks.

desk verdict A useful empirical benchmark for LLM-derived probability priors in Bayesian networks, but missing significance tests and a real memorization confound keep it from being conclusive. read the letter →

arxiv 2505.15918 v2 pith:DEGDZRCO submitted 2025-05-21 cs.CL cs.AI

Aliakbar Nafar , Kristen Brent Venable , Zijun Cui , Parisa Kordjamshidi This is my paper · ORCID

classification cs.CLcs.AI

keywords BayesiannetworksprobabilisticestimationLLMpriorsexpert-drivenconditionalprobabilitytableslow-dataregimeKLdivergencepromptengineering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that large language models can be treated as cheap, always-available stand-ins for domain experts when building Bayesian networks: asked to estimate the conditional probabilities of a node given its parents, they produce distributions closer to ground truth than uniform, random, or token-probability baselines across eighty real-world networks. It then claims that these estimates work as expert priors. Combined with small amounts of data through a pseudocount update, the LLM prior, called Expert-Driven Priors (EDP), yields lower KL divergence than maximum-likelihood estimation or uniform priors, with the largest gains when data is scarce. If true, the result matters because it automates parameterization of Bayesian networks in domains where expert elicitation is costly or unavailable, and it gives a way to get usable probability tables from just a few samples. The paper does not claim to learn the network structure; it assumes the structure is given.

What carries the argument

The load-bearing mechanism is the pairing of a prompting scheme with a pseudocount update. SepState asks the LLM one question per state of a node, conditioned on a fixed parent configuration, has the model reason aloud, and normalizes the returned values so they sum to one; FullDist instead requests the whole tuple in one answer, which the paper finds less accurate in 7 of 8 models. The EDP estimator then treats the normalized LLM output $q_1,\dots,q_m$ as a prior: each $q_i$ is converted into $\alpha q_i$ virtual observations and mixed with empirical counts $c_i$ via $p_i = (\alpha q_i + c_i)/(\alpha + \sum_j c_j)$, with $\alpha$ controlling how much the model is trusted relative to data. The argument depends on evaluating these outputs against published ground-truth conditional probability tables using KL divergence.

What would settle it

Re-run the SepState and EDP evaluations on networks whose source papers appeared after the LLM's training cutoff, or whose node descriptions are paraphrased so no original parameter values are recoverable; if the estimates collapse toward the uniform baseline in that setting, the 'probabilistic knowledge' interpretation would be refuted in favor of memorization.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that LLM-generated conditional probabilities are genuinely informative about real-world event relationships, not just fluent guesses. Across 80 discrete Bayesian networks with known ground-truth parameters, the SepState prompting scheme, which asks one question per state and normalizes the answers, produces joint distributions with lower Bayesian-network KL divergence than the uniform prior, random numbers, and raw next-token probabilities, in every non-reasoning model tested. Separating the estimates into a prior and fusing them with empirical counts as $\alpha q_i$ pseudocounts gives the EDP estimator, which improves on maximum likelihood and on uniform-prior smoothing at small sample sizes and still helps at moderate sizes; the accuracy gain carries over to downstream classification, where EDP matches or beats data-only estimation in most settings and at every data size under a naive-Bayes structure. The authors frame this as establishing the first broad benchmark for LLM probabilistic knowledge extraction, using full distributions over multi-state nodes rather than single confidence numbers.

Load-bearing premise

The load-bearing premise is that the model's probability estimates come from general learned knowledge of the world, not from having memorized the exact parameters of the eighty test networks; the prompts are built from the papers that define those networks.

Editorial extensions

If this is right

With as few as 3 data samples, EDP with an LLM prior achieves lower Bayesian-network KL divergence than maximum likelihood trained on 30 samples and than the LLM's raw SepState estimates.
EDP remains beneficial at moderate sample sizes, such as 1,000 samples, by shrinking variance and improving median KL divergence; maximum likelihood only wins at very large sample sizes under unbiased sampling.
On nine classification datasets, EDP beats or ties MLE in low-resource regimes across both Hill Climbing and naive-Bayes structures, and matches or edges out MLE in 12 of 18 full-data cases.
SepState is the recommended extraction scheme when accuracy matters, while FullDist trades accuracy for fewer queries.
Reasoning-focused models estimate probabilities less well than their non-reasoning counterparts, so more chain-of-thought reasoning does not automatically produce better priors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the LLM's advantage comes from memorizing the papers that define the test networks rather than from general domain knowledge, the EDP gains would shrink for networks and parameter tables published after the model's training cutoff; this is testable by withholding the source papers when building node descriptions.
The same pseudocount framework could let $\alpha$ be calibrated per node, per domain, or from the LLM's own confidence, rather than globally by sample size, potentially improving low-data estimates further.
Because EDP only needs the network structure, it plugs directly into structure-learning pipelines, making fully automatic Bayesian-network construction from text a realistic near-term target.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the take: this paper gives us the first large-scale evaluation of LLM probability elicitation against ground-truth CPTs of 80 real Bayesian networks. The method itself is standard pseudocount smoothing from Zhai and Lafferty, so don't expect a new estimator. The contribution is the scale of the evaluation and the finding that LLM priors beat uniform priors, especially in low-data regimes.

What the paper does well: the evaluation is external, using bnRep ground truth, so the main claim is not circular. They compare against random, uniform, token-probability, and random-variable baselines, and they include multiple LLMs and reasoning models. The downstream classification experiments are a useful addition, and they are honest about the modest and sometimes negative gains on a few datasets. The appendix covers alternative sampling methods and per-state/parent analyses, and code and data are promised.

Where the soft spots are: the statistical reporting is weak. The KL comparisons are boxplots with no paired significance tests or confidence intervals across the 80 networks; "consistently outperforms" is doing a lot of work. The EDP hyperparameter alpha for the KL experiments is described only as proportional to the inverse of sample size, with no exact form or sensitivity analysis; in the classification experiments it is tuned on a dev set. If the advantage over Uniform-# depends on alpha, they need to show that.

The bigger concern is memorization. Appendix A says GPT-4o was used to generate node descriptions directly from the PDFs of the bnRep source papers. Those descriptions go into the prompts, so the model is effectively reminded of the exact paper it's asked to estimate. The LLM(Random) baseline only replaces variable names; it does not control for retrieval of the original CPTs. This weakens the claim that the estimates come from general probabilistic knowledge rather than training-data memorization. A held-out set of BNs not in the training data would settle it, and I'd want that before publishing the central claim as robust.

Who this is for: anyone working on automated BN construction, LLM uncertainty, or low-data probabilistic modeling. It deserves a serious referee, but a referee should push hard on the statistical tests and the memorization confound. I'd send it to review, with the expectation of major revision.

Referee Report

3 major / 5 minor

Summary. The manuscript addresses the problem of parameterizing a Bayesian network (BN) when its structure is given but expert-elicited conditional probability tables (CPTs) are unavailable. The authors propose eliciting probability estimates from LLMs using two prompting schemes (SepState, which queries each state separately, and FullDist, which asks for the full distribution at once), normalizing the outputs, and then optionally fusing them with observed data through an 'Expert-Driven Priors' (EDP) pseudocount update controlled by a hyperparameter alpha. They evaluate on 80 discrete BNs from bnRep against ground-truth parameters, using BN-level and CPT-level KL divergence, and compare against random, uniform, randomized-name, no-context, and token-probability baselines. They also run downstream classification experiments on nine datasets. The central claims are that SepState and FullDist estimates are closer to ground truth than uniform and random baselines, and that EDP improves over MLE and uniform-prior estimates, particularly in low-data regimes.

Significance. If the central claims held, this would be a useful contribution: a reproducible, external-ground-truth benchmark for LLM probabilistic estimation, and a practical recipe for using LLM outputs as Bayesian priors. Strengths include the scale of the evaluation (80 BNs, several LLM families), the use of bnRep ground truth rather than self-reported confidence, the inclusion of multiple baselines, the downstream classification evaluation, and the released code. The main risks are that the apparent probabilistic knowledge may partly reflect memorization of the source papers from which node descriptions are generated, that the EDP advantage is not tied to a fully specified alpha-selection procedure, and that the reported differences lack inferential statistics. These issues do not invalidate the benchmark, but they must be addressed before the performance claims can be accepted.

major comments (3)

[Appendix A and Section 4 (prompt construction)] The prompts used for elicitation contain node descriptions and state definitions generated by GPT-4o directly from the PDFs of the source papers for each bnRep network (Appendix A, Figure 4). If those papers are in the LLM training data, the model may retrieve the original CPTs, so the reported advantage over uniform and random baselines may overstate generalized probabilistic knowledge. The LLM(Random) baseline only randomizes variable names and does not control for this retrieval path. Please add a control in which descriptions are constructed without access to the source PDFs (for example, from node names and state labels only), or evaluate on networks published after the LLM training cutoff, or paraphrase and obfuscate the descriptions; without such a control, the central claim that LLMs possess reusable probabilistic knowledge is not yet established.
[Appendix C and EDP equation in Section 4] The EDP update is pi = (alpha*qi + ci)/(alpha + sum_j cj), but the alpha schedule for the KL-divergence experiments is not fully specified: Appendix C states only that alpha is 'proportional to the inverse of the number of data samples,' without giving the constant or the exact formula. For the classification experiments, alpha is set to 0.5N in the full-data regime and selected from {0.5N, 1.0N, 2.0N} on a development set in the low-data regime. The reader cannot reproduce Figure 3 or assess how much of EDP's advantage comes from the choice of alpha. Please state the exact schedule, report sensitivity of the KL results to alpha, and confirm that alpha was not selected using the ground-truth BN parameters.
[Figures 2-3 and Tables 1-2] The main performance claims are based on comparisons of median KL divergences across 80 BNs and average macro-F1 over nine datasets, but no significance tests, confidence intervals, or paired effect sizes are reported. A Wilcoxon signed-rank test or bootstrap confidence intervals across the 80 networks would make the SepState-vs-uniform and EDP-vs-Uniform-# claims defensible. The classification evidence is also weaker than the text suggests: in Table 1, EDP is below MLE on HouseVotes84 (0.91 vs 0.94) and Puffin (0.93 vs 1.00) in the full-data Hill Climbing regime, and several other entries are ties; the average improvements are 0.01-0.07. Please add inferential statistics and soften the contribution claiming higher classification accuracy accordingly.

minor comments (5)

[Section 5 vs Appendix D (Figure 12)] The main text states that both FullDist and SepState outperform the uniform baseline in all non-reasoning models, but Appendix D reports that under CPT KL divergence FullDist performs worse than the uniform baseline for most LLMs. Please reconcile this discrepancy or qualify the claim to refer specifically to BN-level KL divergence.
[Appendix C (LLM setup)] The appendix says that additional prompts were provided until valid responses were obtained; please report how often retries occurred and whether retried responses were included in the reported cost and quality statistics.
[Table 1] The dataset name 'HouseVotes84' contains a spacing artifact ('HouseV otes84'); please fix the typo.
[Section 6 (EDP with Small Data)] The claim that EDP with 10k samples can still help under biased real-world data is motivated by the classification experiments, but those experiments use 10 or 20 training samples rather than a biased 10k-sample regime; please either add a biased-sampling experiment or rephrase the claim.
[Appendix C (token probability baseline)] The token-probability baseline assigns zero probability to a node state when it is not among the 20 returned candidate strings, which occurs in 12% of cases; please report how sensitive the baseline results are to this cutoff and to the assumption that the returned strings sum to approximately 99%.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: LLM estimates are evaluated against external bnRep ground truth and compared with baselines.

full rationale

The paper's central claim is empirical: LLM-generated conditional probabilities, used alone or as pseudocount priors in EDP, are closer to the fixed ground-truth parameters of 80 bnRep Bayesian networks than uniform, random, token-probability, and data-only baselines. The derivation chain does not define the predicted probabilities in terms of the ground truth: SepState and FullDist elicit raw numeric estimates from LLM text conditioned on node descriptions and then normalize them into distributions, and EDP combines those normalized priors with empirical counts through standard pseudocount smoothing. None of these equations has the target KL divergence or downstream accuracy as an input. Alpha is a hyperparameter selected by a stated heuristic or by dev-set tuning, so it is calibrated rather than derived from first principles, but this does not make the reported result equal to a fitted input by construction. The self-citations (Nafar, Venable, and Kordjamshidi 2024a, 2024b, 2025) appear only in related-work context and are not load-bearing for the evaluation. A genuine validity risk exists in Appendix A, where node descriptions are generated by GPT-4o from the source PDFs of the very networks being evaluated, so LLM outputs may partly reflect memorization rather than general probabilistic knowledge; that concern belongs to external validity, not to circularity of the derivation, and no quoted equation reduces to its own input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central evaluation relies on external ground truth from bnRep, which is appropriate. The main uncharged assumptions are that bnRep CPTs are valid gold references, that the GPT-4o-generated node descriptions are accurate and do not leak the answer, that normalized LLM outputs are genuine probability estimates, and that forward-sampled data mimic real-world scarcity. The only tuned parameter is the EDP prior strength alpha, whose exact value is not fully reported. No new entities are introduced.

free parameters (1)

EDP prior strength alpha = Not fully specified; set proportional to the inverse of the number of data samples in KL experiments, and selected…
Controls how much weight the LLM prior receives relative to empirical counts. The exact formula for the main KL experiments is not reported, and the classification value is tuned on data, so the results depend on this choice.

assumptions (5)

standard math The BN KL divergence decomposes into a weighted sum of per-CPT KL divergences for two BNs with the same DAG.
Appendix B derives this identity; it is standard and not in question.
domain assumption bnRep ground-truth CPTs are valid gold references for probability estimation.
All KL divergence scores are computed against these tables; if they are arbitrary or synthetic, the rankings could change. Invoked throughout Section 5.
ad hoc to paper GPT-4o-generated node descriptions are semantically accurate and do not leak ground-truth probabilities into the prompts.
Appendix A describes automatic description extraction from source PDFs with manual fixes; there is no check for memorization or information leakage.
domain assumption Normalizing raw LLM numeric outputs yields a valid conditional probability distribution.
Section 4 divides each raw value by the sum of the states; this assumes the relative proportions are meaningful.
domain assumption Forward sampling from ground-truth BNs produces data that resembles realistic data scarcity and bias patterns.
Sampled data are used for MLE, Uniform-# and EDP-#; real-world missingness and sampling bias could change the comparisons.

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Cite this review

Pith. "Pith review of Extracting Probabilistic Knowledge from Large Language Models for Bayesian Network Parameterization." pith.science (2026). https://pith.science/paper/DEGDZRCO

@misc{pith2026250515918,
  author       = {Pith},
  title        = {Pith review of: Extracting Probabilistic Knowledge from Large Language Models for Bayesian Network Parameterization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DEGDZRCO}},
  note         = {Machine review of arXiv:2505.15918}
}

read the original abstract

In this work, we evaluate the potential of Large Language Models (LLMs) in building Bayesian Networks (BNs) by approximating domain expert priors. LLMs have demonstrated potential as factual knowledge bases; however, their capability to generate probabilistic knowledge about real-world events remains understudied. We explore utilizing the probabilistic knowledge inherent in LLMs to derive probability estimates for statements regarding events and their relationships within a BN. Using LLMs in this context allows for the parameterization of BNs, enabling probabilistic modeling within specific domains. Our experiments on eighty publicly available Bayesian Networks, from healthcare to finance, demonstrate that querying LLMs about the conditional probabilities of events provides meaningful results when compared to baselines, including random and uniform distributions, as well as approaches based on next-token generation probabilities. We explore how these LLM-derived distributions can serve as expert priors to refine distributions extracted from data, especially when data is scarce. Overall, this work introduces a promising strategy for automatically constructing Bayesian Networks by combining probabilistic knowledge extracted from LLMs with real-world data. Additionally, we establish the first comprehensive baseline for assessing LLM performance in extracting probabilistic knowledge.

Figures

Figures reproduced from arXiv: 2505.15918 by the authors.

**Figure 1.** Two-stage parameterization pipeline. SepState (Left Panel): For each parent configuration, the LLM is prompted with natural-language descriptions of the node and its parents and queried once per state. The answers are subsequently normalized (z −1 ) into a valid conditional distribution, e.g., (75%, 25%). EDP (Right Panel): The LLM-derived prior distribution is translated into pseudocounts and fused with empirical … view at source ↗

**Figure 2.** Boxplot showing the distribution of BN KL divergence values across 80 unique BNs for various models, sorted by [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Boxplot of distribution of BN KL divergence over 80 networks, contrasting models with GPT-4o priors (EDP-#), [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (22 more)

**Figure 4.** Figure 4: Example of node descriptions and state extractions [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: KL divergence decomposition into local components for Bayesian networks. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Prompt templates used for eliciting probabilistic [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Boxplot of distribution of BN KL divergence over 80 networks, contrasting models with DeepSeek-V3 priors (EDP-#), [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: Boxplot of distribution of BN KL divergence over 80 networks, contrasting models with Gemini-pro 1.5 priors [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

**Figure 9.** Figure 9: Boxplot of distribution of BN KL divergence over 80 networks, contrasting models with Claude 3.5 Sonnet priors [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

**Figure 10.** Figure 10: Boxplot of distribution of BN KL divergence over 80 networks, contrasting models with GPT-4o priors (EDP-#), [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: Boxplot of distribution of BN KL divergence over 80 networks, contrasting models with GPT-4o priors (EDP-#), [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: Boxplot showing the distribution of CPT KL divergence values across 80 unique BNs for various models, sorted by [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 13.** Figure 13: CPT KL divergence for nodes with 2 states. [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]

**Figure 14.** Figure 14: CPT KL divergence for nodes with 3 states. [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]

**Figure 15.** Figure 15: CPT KL divergence for nodes with 4 states. [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]

**Figure 16.** Figure 16: CPT KL divergence for nodes with 5 states. [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]

**Figure 17.** Figure 17: CPT KL divergence for nodes with 6 states. [PITH_FULL_IMAGE:figures/full_fig_p021_17.png]

**Figure 18.** Figure 18: CPT KL divergence for nodes with 0 parents. [PITH_FULL_IMAGE:figures/full_fig_p021_18.png]

**Figure 19.** Figure 19: CPT KL divergence for nodes with 1 parents. [PITH_FULL_IMAGE:figures/full_fig_p022_19.png]

**Figure 20.** Figure 20: CPT KL divergence for nodes with 2 parents. [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]

**Figure 21.** Figure 21: CPT KL divergence for nodes with 3 parents. [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]

**Figure 22.** Figure 22: CPT KL divergence for nodes with 4 parents. [PITH_FULL_IMAGE:figures/full_fig_p023_22.png]

**Figure 23.** Figure 23: CPT KL divergence for nodes with 5 parents. [PITH_FULL_IMAGE:figures/full_fig_p024_23.png]

**Figure 24.** Figure 24: CPT KL divergence for nodes with 6 parents. [PITH_FULL_IMAGE:figures/full_fig_p024_24.png]

**Figure 25.** Figure 25: CPT KL divergence for nodes with 7 parents. [PITH_FULL_IMAGE:figures/full_fig_p025_25.png]

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Forward citations

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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