Dual Traits in Probabilistic Reasoning of Large Language Models

Huaxia Rui; Shenxiong Li

REVIEW 3 major objections 4 minor 30 references

Dual Traits in Probabilistic Reasoning of Large Language Models

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read State-of-the-art LLMs judge probabilities by Bayes' rule or by similarity, depending on the prompt.

desk verdict A strong semi-structured experiment supports the dual-mode claim, but the unstructured test's circular diagnosticity benchmark needs external validation. read the letter →

arxiv 2412.11009 v1 pith:KX2QI7WJ submitted 2024-12-15 cs.AI cs.CLcs.CY

Shenxiong Li , Huaxia Rui This is my paper · ORCID

classification cs.AIcs.CLcs.CY

keywords largelanguagemodelsBayesianreasoningrepresentativenessheuristicbase-rateneglectposteriorprobabilitysimilarityjudgmentpromptengineeringreinforcementlearningfromhumanfeedback

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

Large language models do not sit at a single point on a biased-versus-unbiased scale. This paper claims that their posterior judgments $P(H|E)$ are produced by two coexisting modes: a normative mode that follows Bayes' rule and a representativeness mode that answers by how similar the evidence is to a typical member of the class. In a test that varied only the base rate, state-of-the-art models produced normative posterior-odds ratios near 9 for a neutral description but near 1 for representative descriptions, meaning the base rate was ignored exactly when the evidence was stereotypical. In a second test with no given base rate or diagnosticity, judged posteriors correlated strongly with similarity (Pearson $r \approx 0.9$) and negatively with base rates, even though the models' own inverse-probability judgments implied the evidence was nearly non-diagnostic. The paper also shows that a single wording change can switch representative judgments to Bayesian ones, and conjectures that the dual trait comes from contrastive losses in reinforcement learning from human feedback.

What carries the argument

The load-bearing object is a dual-mode posterior function $f(H,E)$ whose normative component $f_{\mathrm{norm}}$ follows Bayes' rule and whose representativeness component $f_{\mathrm{rep}}$ equates posterior probability with similarity. The key identity that separates the two modes is the base-rate odds ratio: because evidence diagnosticity is independent of the base rate, the ratio of posterior odds across the 75% and 25% base-rate conditions must be exactly 9 under $f_{\mathrm{norm}}$, and drifts toward 1 as $f_{\mathrm{rep}}$ takes over. In the unstructured test, the auxiliary object is diagnosticity $D(H,E)=P(E|H)/P(E|\neg H)$ computed from the model's own inverse-probability judgments; values near 1 license the benchmark that normative posteriors equal priors, so a posterior that tracks similarity rather than priors implicates $f_{\mathrm{rep}}$. The paper's conjectured origin mechanism is the binary ranking loss used in reward-model training, which the authors read as a contrastive loss that amplifies distinguishing features while ignoring class frequencies.

What would settle it

Ask independent human judges to estimate $P(E|H)$ and $P(E|\neg H)$ for the Adam sketch across the three fields, and compare those externally elicited diagnosticity values with the models' values. If the external $D(H,E)$ values are near 1 while the LLMs' posteriors still track similarity and move away from priors, the dual-mode and base-rate-neglect claims for the unstructured test are confirmed; if the external $D(H,E)$ values are large, the experiment cannot distinguish correct diagnosticity-based updating from representativeness.

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Extended reading notes

Core claim

The paper's central claim is that posterior judgment in state-of-the-art LLMs is driven by two distinct prediction functions: $f_{\mathrm{norm}}$, which applies Bayes' rule by combining base rates with evidence diagnosticity, and $f_{\mathrm{rep}}$, which maps similarity between the evidence and a class prototype directly to probability. The coexistence of these modes is established with an odds-ratio design: when the base rate is 75% versus 25%, any normative judge must give posterior-odds ratio $O(B_h)/O(B_\ell)=9$ regardless of the description, whereas representative descriptions push all tested models to ratios near 1 even though the base rates are stated in the prompt. In the unstructured test, where the Adam sketch is judged across three fields with different priors, posteriors are negatively correlated with base rates and positively correlated with similarity, while the models' own inverse-probability estimates imply diagnosticity $D(H,E)$ near 1, so normative posteriors should track priors. The paper further shows that prompting a model to 'compute' rather than 'indicate' restores Bayesian odds ratios in the semi-structured setting, but that iterative Bayes-rule prompting in the unstructured setting does not eliminate similarity-driven judgment, and that models recall base rates from memory poorly.

Load-bearing premise

The unstructured test's conclusion that normative posteriors should equal priors rests on the premise that the Adam description is non-diagnostic, and that premise is supported only by diagnosticity values computed from the same LLMs' inverse-probability judgments, which themselves correlate almost perfectly with similarity.

Editorial extensions

If this is right

LLMs can compute normatively when all ingredients of Bayes' rule are present or explicitly cued, so poor accuracy on unconstrained questions does not by itself demonstrate absence of normative competence.
Base-rate neglect is mode-dependent rather than universal: representative descriptions suppress sensitivity to base rates even when the base rates are stated in the prompt.
A minimal wording change from 'indicate' to 'compute' can flip a model from the representativeness mode to the normative mode in the semi-structured setting, making surface prompt features a strong lever on mode selection.
In unstructured settings, prompting the model to apply Bayes' rule does not eliminate similarity-driven posterior judgments, and models recall base rates from memory poorly, so deployed systems should supply priors explicitly.
If the paper's conjecture about contrastive loss is right, the reward-model stage of reinforcement learning from human feedback may itself entrench representativeness, meaning bias mitigation may have to alter training objectives rather than only prompts.

Reading between the lines

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

Because the non-diagnosticity of the Adam sketch is established with the same models whose similarity-driven reasoning is under investigation, the unstructured test's base-rate-neglect result should be read as conditional; an externally validated set of inverse probabilities would settle whether the benchmark is contaminated.
The odds-ratio test used in the semi-structured experiment is cheap and model-agnostic: ask any model for posteriors under 75% and 25% base rates and compare the ratio of posterior odds to 9, with no inverse probabilities needed.
If the contrastive-loss conjecture is correct, it predicts a testable ordering: models trained with preference-optimization losses that ignore class frequencies should show lower odds ratios on representative descriptions than models trained on next-token prediction or on objectives that include base rates.
Wording sensitivity implies that mode selection may be triggered by token-level cues that invoke arithmetic routines, so interventions that force explicit step-by-step calculation before output may generalize better across prompts than substituting a single verb.

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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

The paper does something genuinely useful: it tests LLMs on posterior probability judgment with three levels of information, and finds a shift from near-normative Bayesian behavior (when all numbers are given) to representativeness-driven behavior (when the prompt omits diagnosticity and pushes toward a stereotype). The semi-structured 'Jason' test is the strongest piece — a fixed 9:1 odds ratio benchmark that makes the prediction unambiguous, and a nice control where LLMs mostly track base rates on an uncharacteristic description but ignore them on representative ones. The single-word prompt reversal ('indicate' → 'compute') is a clean and clear demonstration that the bias is not hard-wired. Those results alone justify the paper's attention.

The unstructured test is where I'd push back. The claim that the Adam sketch is non-diagnostic rests on D values obtained from the same five LLMs whose inverse probabilities correlate ~0.9 with similarity. That is partially circular: if the D judgments themselves are products of the representativeness mechanism, the benchmark is biased. The stress-test note quantifies the issue for gpt-4o: the observed posterior odds require a D(A)/D(C) ratio around 15, but the elicited D values give roughly 0.73. That does not kill the paper, because the semi-structured test already shows base-rate neglect in representative contexts, but the unstructured test's negative prior–posterior correlation loses its clean normative contrast. The small number of rotations (7 per model) and the missing error bars on the odds ratios only add to that concern.

The RLHF contrastive-loss conjecture is clearly labeled as speculative, and the paper ships code and data, which I count in its favor. The writing is straightforward and the citations to the base-rate literature are appropriate.

Bottom line: the central dual-mode finding is plausible and worth building on, but the unstructured test needs an external diagnosticity check — ideally human or independent-likelihood judgments — and fuller uncertainty reporting. I'd send this to peer review; it deserves a serious referee, with the expectation of revision focused on the unstructured test.

Referee Report

3 major / 4 minor

Summary. This paper reports three experiments on how large language models (LLMs) judge posterior probabilities P(H|E). In the structured test, where base rates and likelihoods are fully supplied, state-of-the-art LLMs solve textbook Bayesian problems at high accuracy. In the semi-structured test, only base rates are supplied; for a description high in representativeness, all tested LLMs produce posterior odds near 1 instead of the normative odds ratio of 9, while for a non-representative description they respond to the base-rate manipulation. Replacing the word 'indicate' by 'compute' in the prompt restores a near-9 odds ratio for GPT-4o. In the unstructured test, with no Bayesian information supplied, posterior judgments correlate negatively with prior probabilities and positively with similarity judgments; the authors infer base-rate neglect by using the same models' inverse-probability estimates to argue that the stimulus is non-diagnostic. The paper concludes that LLMs exhibit coexisting normative and representativeness-based modes and conjectures that this duality stems from the contrastive-style loss used in reinforcement learning from human feedback.

Significance. The semi-structured test is a clean, externally normed demonstration that LLM posterior judgments can be insensitive to base rates while being strongly driven by representativeness, and the prompt-reversal control shows that a single wording change can shift behavior toward Bayesian combination. If this result holds, the paper's central 'dual modes' claim would be an important advance over simple bias/no-bias characterizations. The authors also make code and data available and use repeated sampling with predetermined questioning rounds to stabilize estimates, which strengthens the empirical contribution. The main limitation is that the unstructured test's normative benchmark is derived from the same models under test, whose likelihood judgments correlate almost perfectly with similarity; that test therefore does not independently establish base-rate neglect. This tempers the breadth of the claims, but the core semi-structured evidence remains defensible.

major comments (3)

[Unstructured Test, Extended Data Tables A.2 and A.3] The claim that normative posteriors in the unstructured test should track priors rests on the assertion that the Adam sketch is non-diagnostic (D≈1). This assertion is validated only by D(H,E) values in Table A.2, which are computed from P(E|H) and P(E|¬H) elicited from the same five LLMs. Table A.3 shows these inverse probabilities correlate almost perfectly with similarity (Pearson 0.92–0.98), so the 'non-diagnosticity' benchmark is generated by the same representativeness mechanism the experiment is designed to detect. The quantitative gap is large: for gpt-4o in the single-field rotation, the prior odds P(A)/P(C)=0.02/0.13≈0.15 while the posterior odds P(A|E)/P(C|E)=0.44/0.19≈2.3, requiring D(A)/D(C)≈15 to be normatively consistent; Table A.2 reports D(E,A)=1.09 and D(E,C)=1.50, a ratio of 0.73. If the sketch is actually diagnostic for field A, the negative prior–posterior correlation is not evidence of base-rate neglect. Please add an external diagnosticity check (e.g., human likelihood ratings) or reframe the unstructured-test conclusions as descriptive evidence of similarity-based judgment without claiming a normative violation.
[Methods, Rotational Design; Table 2] The correlation evidence in the unstructured test is based on only 7 rotations per model. With n=7, a two-sided p-value below 0.001 is highly dependent on the exact permutation set, and the current footnote 'two-sided Fisher's p-value' is ambiguous because Fisher's z-transformation is monotonic in r and does not by itself define a nonparametric test for n=7. Please report the exact permutation p-values or bootstrap confidence intervals for the correlations in Tables 2 and A.5, and state how many permutations were used. The small sample also means the 'prior vs. posterior' correlation could be driven by a single rotation; please report the raw means for all seven rotations or a leave-one-rotation-out sensitivity analysis.
[Prompt Engineering; Extended Data Table A.4; SI Section 2.4] The striking finding that replacing 'indicate' with 'compute' reverses base-rate neglect is demonstrated only for GPT-4o. The semi-structured test's across-model claim is established, but the causal 'single word can trigger the normative mode' claim should be either replicated on at least a few other state-of-the-art models or explicitly qualified as a single-model demonstration. As written, the Discussion and Abstract generalize this prompt-reversal control beyond the evidence.

minor comments (4)

[Structured Test, SI Section 2.1] The accuracy interval (0.40, 0.43) for Q7, Q8, and Q10–Q12 is sensible given the exact Bayesian answer of about 0.4138, but please state explicitly that this interval is an analytic rounding convention rather than a post hoc fitted tolerance, and note whether it was fixed before data collection.
[Supplementary Information Section 3] The sentence introducing the simultaneous confidence intervals contains '[see, e.g. ? , §4.4]' with a missing citation; please complete or remove the reference.
[Table 1] The human comparison column is based on different base rates (0.7/0.3, with the note stating a normative ratio of about 5.44), so the direct comparison with the LLM rows' 9:1 normative benchmark is not clean; please present a normalized measure such as the ratio of posterior odds divided by the prior odds so that human and LLM rows are on the same scale.
[Throughout] There are several typographical and formatting issues, including 'T able' spacing throughout, 'Inverse Probabiliy' in SI Table A.2 headers, and 'It has been showed' in the Discussion (should be 'It has been shown').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central claims rest on externally benchmarked structured/semi-structured tests and independent posterior-similarity correlations, not on fitted parameters or load-bearing self-citations.

full rationale

The paper's central derivation chain is not circular. The structured test is scored against externally known correct answers, and the semi-structured test derives a normative posterior-odds ratio of 9 by canceling the unknown evidence diagnosticity D(H,E) between the two base-rate conditions, so its benchmark does not depend on any LLM-generated quantity. The unstructured test's D values are elicited from the same LLMs, but they function as a coherence benchmark: the paper compares the LLMs' posteriors with their own stated priors and inverse probabilities, rather than fitting a parameter and then predicting the same data. Even if those inverse-probability estimates are themselves similarity-driven (Table A.3 shows high inverse-vs-similarity correlations), this is a validity or calibration concern about how to interpret the unstructured test, not a reduction of the conclusion to its inputs by construction. The posterior-similarity correlation is an independently elicited empirical correlation, and the representative-mode claim is also supported by the semi-structured test's robust prior-odds-ratio result, which is external to the model's judgments. The paper contains no load-bearing self-citations; all cited prior work is external psychology and machine-learning literature. The Discussion's contrastive-loss conjecture is explicitly labeled a conjecture and is not used to derive the experimental conclusions. The Methods section candidly discusses metalinguistic-prompt and data-contamination limitations, which further supports that the authors are not hiding a definitional equivalence. No circular step meeting the quoting-and-reduction standard was found.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The empirical claims rely on standard statistical assumptions and on the external validity of the prompts as tests of Bayesian reasoning. The only hand-set numerical choice is the accuracy tolerance interval. The paper introduces no new theoretical entities; the normative and representativeness modes are borrowed from the psychology literature.

free parameters (1)

accuracy tolerance interval for Q7/Q8/Q10-Q12 = (0.40, 0.43)
The true Bayesian answers for these questions are approximately 0.4138; the paper defines any answer in this interval as correct, a hand-chosen tolerance that affects reported accuracy rates.

assumptions (4)

domain assumption LLM responses are i.i.d. samples from a fixed response distribution when queried repeatedly with memory reset.
The questioning-round methodology and confidence intervals rely on this; the paper resets context and assumes API responses do not feed back into the model during the short experimental window.
standard math The evidence's diagnosticity D(H,E) is independent of the base rate B in the semi-structured test; the same description is drawn from the same evidence distribution under H in both base-rate conditions.
This assumption justifies the fixed 9:1 posterior-odds ratio as the normative benchmark in the Jason test.
domain assumption The Adam description is non-diagnostic of graduate field in the real world; a fully rational judge would have posteriors close to priors.
The paper asserts the description is representative but not diagnostic; the only quantitative support is the LLMs' own D near 1 judgments, which are correlated with similarity.
domain assumption Human benchmark results from the cited psychology literature are validly interpretable in the comparison table.
Human averages are taken from prior studies with different base rates (0.7/0.3), and the paper adjusts the normative ratio to 5.44 accordingly.

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Cite this review

Pith. "Pith review of Dual Traits in Probabilistic Reasoning of Large Language Models." pith.science (2026). https://pith.science/paper/KX2QI7WJ

@misc{pith2026241211009,
  author       = {Pith},
  title        = {Pith review of: Dual Traits in Probabilistic Reasoning of Large Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KX2QI7WJ}},
  note         = {Machine review of arXiv:2412.11009}
}

read the original abstract

We conducted three experiments to investigate how large language models (LLMs) evaluate posterior probabilities. Our results reveal the coexistence of two modes in posterior judgment among state-of-the-art models: a normative mode, which adheres to Bayes' rule, and a representative-based mode, which relies on similarity -- paralleling human System 1 and System 2 thinking. Additionally, we observed that LLMs struggle to recall base rate information from their memory, and developing prompt engineering strategies to mitigate representative-based judgment may be challenging. We further conjecture that the dual modes of judgment may be a result of the contrastive loss function employed in reinforcement learning from human feedback. Our findings underscore the potential direction for reducing cognitive biases in LLMs and the necessity for cautious deployment of LLMs in critical areas.

Discussion (0). Continue with ORCID to comment.

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For each of the following ﬁelds, please provide your best e stimates of the posterior conditional probability that Ada m was enrolled in that ﬁeld given his personality sketch
[28]

For each of the following ﬁelds, please compute the poster ior conditional probability that Adam was enrolled in that ﬁ eld given his personality sketch
[29]

For each of the following ﬁelds, denote H to be the hypothesis that Adam was enrolled in that ﬁeld

Let E denote the personality sketch of Adam. For each of the following ﬁelds, denote H to be the hypothesis that Adam was enrolled in that ﬁeld. Then, please compute the posterior co nditional probability P(H | E) using Bayes’ rule
[30]

For each of the following ﬁelds, denote H to be the hypothesis that Adam was enrolled in that ﬁeld

Let E denote the personality sketch of Adam. For each of the following ﬁelds, denote H to be the hypothesis that Adam was enrolled in that ﬁeld. Now , please ﬁrstly judge P(H), P(E | H) and P(E | \neg H), and then use them to compute P(H | E) through Bayes rule. 3 Questioning ...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

URL https://arxiv .org/abs/1902.09229

Arora S, Khandeparkar H, Khodak M, Plevrakis O, Saunshi N (2019) A theoretical analysis of contrastive unsu- pervised representation learning. URL https://arxiv .org/abs/1902.09229

arXiv 2019

[2] [2]

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