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REVIEW 3 major objections 5 minor 42 references

Recovering Event Probabilities from Large Language Model Embeddings via Axiomatic Constraints

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A sign-flipped latent variable recovers event probabilities from LLM embeddings that add to one.

desk verdict A clever proof-of-concept for extracting coherent probabilities from LLM embeddings via a sign-flip VAE, held back by an under-tested single-latent assumption and some reporting ambiguities. read the letter →

arxiv 2505.07883 v1 pith:LW3OCMMU submitted 2025-05-10 cs.CL cs.AI

Jian-Qiao Zhu , Haijiang Yan , Thomas L. Griffiths This is my paper · ORCID
classification cs.CLcs.AI
keywords eventprobabilitiesLLMembeddingscoherencevariationalautoencoderlog-oddsrepresentationaxiomaticconstraintscomplementaryeventsdiceprobabilitycalibration
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

This paper asks whether coherent probabilities can be recovered from LLM embeddings even when the probabilities the model writes in text violate basic axioms. It claims yes: training a two-step variational autoencoder on embeddings of events and their complements, with complements generated by flipping the sign of one latent variable, makes probability emerge as log-odds in that dimension. Without ever being shown true probabilities, the recovered probabilities on held-out dice questions satisfy the complement rule far better than the model's text judgments and track the true probabilities. The broader point is that embedding space contains probability information that the model's own verbal responses do not faithfully report.

What carries the argument

The load-bearing object is the first latent variable $z^{(1)}$ of a $\beta$-weighted variational autoencoder ($\beta$-VAE) trained directly on LLM embeddings. In Step 1 the VAE reconstructs embeddings from a ten-dimensional Gaussian latent space; in Step 2 it generates the complement embedding by applying $T(z)=[-z^{(1)}, z^{(-1)}]$ to the sampled latent vector. The centered isotropic Gaussian prior supplies the axiom: in log-odds space (i.e., $\log(P/(1-P))$) the additive rule for complements is that log-odds sum to zero, so the sign-flip transformation combined with the logistic function $P=e^{z_1}/(1+e^{z_1})$ turns the latent operation into the probability-theoretic constraint.

What would settle it

Construct a held-out set where complements are phrased without the word 'not' (for dice, 'rolling a 1, 2, 3, 4, or 5' versus 'rolling a 6') or use lexically unrelated complement pairs; then check whether the sign-flipped latent still recovers the true probabilities. If correlation with true probabilities drops toward zero while the text-judged probabilities stay accurate, the recovered latent was tracking the surface marker, not probability.

Watch

Extended reading notes

Core claim

The central discovery is that enforcing the complement rule in latent space is enough to recover coherent probability estimates. In the trained encoder-decoder, the first latent variable $z^{(1)}$ behaves as log-odds: the complement embedding is generated by $T(z)=[-z^{(1)}, z^{(-1)}]$, and the centered Gaussian prior makes the log-odds of complementary events sum to zero; converting with $P=e^{z_1}/(1+e^{z_1})$ then enforces $P(A)+P(\neg A)=1$. On a held-out test set of 480 dice events, probabilities recovered from Gemma-2-9b-instruct embeddings have incoherence 0.0383 versus 0.1366 for text judgments, correlate with true probabilities at $r=0.7328$, and reach MSE=0.1014. The same procedure improves coherence for Llama-3.1-8b-instruct, although with lower accuracy, suggesting that recovered quality tracks the model's underlying capability.

Load-bearing premise

The entire method rests on the assumption that all probability-relevant information in an embedding is concentrated in a single latent dimension that flips sign when the event is complemented, and that this dimension is truly log-odds rather than a surface feature such as the word 'not'.

Editorial extensions

If this is right

  • Coherent probabilities can be extracted from embeddings without true labels, opening an unsupervised route to calibrating LLM uncertainty in dice-like decision problems.
  • The method splits the latent space into interpretable dimensions: the modified latent tracks true probability, while others encode prompt features such as number of rolls, sum-versus-single-roll, and comparison type.
  • Recovery quality is tied to the source model: Gemma's recovered probabilities beat its own text judgments on coherence and match them on accuracy, while Llama's recovered probabilities improve coherence but remain less accurate, implying the internal representation's quality limits the method.
  • The ablation implies that predicting complement embeddings (Step 2) is necessary; without it a standard beta-VAE does not single out a probability-specialised latent variable, and the recovered probabilities are less coherent and less accurate.
  • A linear probe trained on true probabilities excels on training events but fails on held-out events, so the axiomatic unsupervised method generalises where supervised probing does not.

Reading between the lines

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

  • Editorial inference: if the one-dimensional log-odds encoding generalises beyond dice, the same sign-flip constraint could be applied to any paired event embeddings, effectively giving a coherence regulariser for downstream decision systems; the paper only demonstrates dice events.
  • Editorial inference: the Gemma versus Llama comparison suggests a testable diagnostic—use the VAE's recovered coherence as a proxy for the model's internal probabilistic calibration, independent of its text outputs.
  • Editorial inference: because $z^{(1)}$ appears to encode log-odds, intervening on it (e.g., flipping its sign while decoding) might change judged probabilities for complements, providing a causal route the paper does not test.
  • Editorial inference: the main risk is that $z^{(1)}$ captures the surface word 'not' rather than probability; a direct test is to use complements expressed without 'not' or with unrelated wording, and if the recovered probabilities collapse, the claim of probability encoding fails.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes an unsupervised variational-autoencoder method for recovering event probabilities from LLM embeddings. Given embeddings e and ¬e for complementary events, a VAE is trained to reconstruct e and, in a second step, to predict ¬e by sign-flipping the first latent variable (T(z)=[-z^(1), z^(-1)]). The first latent variable is interpreted as log odds and converted to a probability via the logistic function, so that the Gaussian prior plus the sign flip enforces P(A)+P(¬A)=1. On dice events (1,728 training and 480 held-out questions) using Gemma-2-9b-instruct embeddings, the recovered probabilities have test incoherence 0.038 versus 0.137 for raw judged probabilities, correlate with true probabilities at r=0.73, and have MSE statistically indistinguishable from judged probabilities. An ablation that removes the complement-prediction step yields worse coherence, and a linear probe trained on true probabilities fails to generalize to held-out events. A replication with Llama-3.1-8b-instruct is reported in Appendix I.

Significance. If the central claim survives additional scrutiny, this is a useful contribution to interpretability and uncertainty estimation for large language models: it shows that a coherence prior can be injected through a latent-space symmetry rather than by post-hoc normalization, and that the resulting latent variable is partially interpretable. The paper has real strengths: a held-out test set with known ground-truth probabilities, an ablation study, a linear-probe comparison, and a second-model replication. The main risk is that the single sign-flip assumption is not directly validated, and the absolute calibration of the recovered probabilities depends on an unexplained temperature parameter. These are load-bearing issues rather than cosmetic ones, so the paper needs revision before the claims can be accepted at face value.

major comments (3)
  1. [Section 3, Eq. (3)] The reduction p(¬e|z,e) ≈ p(¬e|T(z)) with T(z)=[-z^(1), z^(-1)] is the load-bearing step of the method, but the paper provides no direct evidence that the decoder actually predicts held-out complementary embeddings through this single sign-flip. Because the dice template has essentially one probability-relevant scalar and many memorizable surface features (e.g., the word 'not' or numeric tokens), a latent z^(1) encoding any monotone confound correlated with probability would satisfy the training objective. Please report a quantitative measure of how well p_θ(¬e|T(z)) predicts held-out complementary embeddings, and test the sign-flip assumption on at least one non-dice domain or with templates that decorrelate surface features from probability.
  2. [Section 4, temperature parameter] The scale of z^(1) is determined only by the VAE prior and reconstruction losses; converting z^(1) to a probability via the logistic function with 'temperature=5' introduces a free parameter that governs absolute probability calibration. Since the true probability is never observed during training, the reported MSE and correlation for P_recovered depend on this arbitrary scaling. Please justify the temperature value, report sensitivity to it, and either calibrate it on a validation split or derive it from the Gaussian prior and the trained decoder.
  3. [Section 4, Table 1 and Abstract] The abstract's claim that recovered probabilities 'align closely with the true probabilities' overstates the accuracy result: on the test set, P_recovered has MSE 0.1014, which is numerically worse than P_judged's 0.0927 and not statistically different (t(959)=-1.58, p=.12). Moreover, normalized P_judged has much lower incoherence (0.0021) than P_recovered (0.0383). The precise contribution is therefore improved coherence relative to raw P_judged, with accuracy comparable to direct judgments, not improved accuracy. Please reframe the claims accordingly and include normalized P_judged in the headline comparison.
minor comments (5)
  1. [Table 1] The 95% confidence interval for normalized P_judged on the test set is [-0.0020, 0.0062], but incoherence is a nonnegative quantity by definition (Eq. 8); this interval likely comes from a normal approximation and should be replaced with a bootstrap or other method that respects the nonnegative support.
  2. [Section 4, temperature description] Please state explicitly whether 'temperature=5' divides z^(1) inside the sigmoid or scales the logit in Eq. (6); the current text says 'we applied the same value of 5 for temperature' without a formula, which makes the exact mapping ambiguous.
  3. [References, [14]] The reference to the Gemma team paper contains a typo ('none Gemma Team'); please correct this.
  4. [Appendix I] There is a typo in the text: 'P_judged from the Llamma model' should be 'Llama model'.
  5. [Appendix F, Table 3] For layers 25 and earlier the correlations between recovered and true probabilities become strongly negative, but the paper does not comment on why this happens; a brief explanation or discussion would improve confidence in the method's stability across layers.

Circularity Check

2 steps flagged · score 4.0 of 10

Coherence claim is self-definitional: P_recovered is defined as sigmoid(z1) and the complement via sign-flip, so P(A)+P(not A)=1 is a mathematical identity; true-probability alignment remains an independent empirical result.

  1. self definitional [Section 3, Equations (6)-(7); implemented in Section 4, 'Recovered probabilities'.]
    "Finally, the log odds recorded in the modified latent variable, z(1), are converted back to the probability scale: P_recovered = e^{z(1)}/(1+e^{z(1)}) (6) By flipping the sign of z(1), we enforce that the recovered probabilities satisfy the additive rule where the probabilities of complementary events must add up to 1: P_recovered + (1−P_recovered) = e^{z(1)}/(1+e^{z(1)}) + e^{−z(1)}/(1+e^{−z(1)}) (7)"

    The paper presents the coherence of P_recovered as a recovered property of LLM embeddings, but Equation (7) is the logistic-function identity sigma(x)+sigma(-x)=1. Because P_recovered is defined as sigmoid(z(1)) and the complementary event's probability is defined via the sign-flipped T(z), the additivity constraint is true by construction for any value of z(1). The reported improvement in incoherence over P_judged is therefore a formal consequence of the estimator's definition, not an empirical discovery about the embeddings. The independent content of the paper—the correlation of z(1) with true dice probabilities—does not follow from this identity, but the headline coherence claim does reduce to definition.

  2. self definitional [Section 3, Equation (3); Section 4, Figure 3 and surrounding text.]
    "where the modification operation consists of negating the first latent variable while preserving all others: T(z) = [−z^(1), z^(−1)]. ... Only the modified latent variable 1 exhibits a negative relationship."

    Step 2's training objective is to predict the complementary embedding from the latent vector with z(1) negated and all other latent variables unchanged. The finding that z(1) is the only latent variable with a negative relationship between complementary events is therefore induced by the chosen modification operator rather than being an independent confirmation that event probability is encoded in that dimension. The paper states that this 'suggests that the relationship of complementary events between e and ¬e has been successfully distilled into this specific latent variable,' but the distillation target—sign-flipping z(1)—was specified in advance in Equation (3).

full rationale

The derivation chain is mostly self-contained against external benchmarks: the VAE objective never observes P_true, the test set consists of held-out dice events, the models are open-weight, and the observed correlation between the recovered latent and true probabilities is a non-trivial empirical result. The main circularity is that the paper's central coherence claim is enforced by the sigmoid/sign-flip construction in Equations (6)-(7): any scalar z(1) mapped through the logistic function and complemented by sign-flipping automatically satisfies P + (1-P) = 1. Similarly, the choice to modify only z(1) in Step 2 makes it expected that z(1) is the latent that distinguishes complements. These are real but partial circularities: the accuracy claims against true probabilities retain independent content. The hand-set temperature value of 5 (Section 4) also affects the probability scale and is not derived from the axioms, but there is no evidence it was fitted to the test set, so it is treated as a calibration caveat rather than a circular step. Overall the paper has non-trivial empirical content, but the coherence portion of the headline reduces to definition.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method adds no new physical or conceptual entities. Its load-bearing assumptions are the single-dimension sign-flip structure and the recoverability of probability from embeddings.

free parameters (3)
  • beta (KL weight) = 5
    Scales the KL divergence in the beta-VAE objective to emphasize the Gaussian prior; chosen by hand with no sensitivity analysis. It affects how strongly the latent is regularized and therefore the calibration of recovered probabilities.
  • temperature for log-odds conversion = 5 (ambiguous)
    The text says 'we applied the same value of 5 for temperature' when converting log odds to probabilities, but Equation (6) shows no temperature. If applied, it changes the probability scale and is a free parameter.
  • latent dimension k = 10
    The number of latent variables was set to 10 by hand; the method assumes the probability lives in the first dimension.
assumptions (4)
  • ad hoc to paper The relationship between complementary embeddings is fully captured by negating one latent variable (Eq. 3).
    This is the central modeling assumption; no derivation from LLM properties is given.
  • domain assumption LLM embeddings encode event probabilities in a recoverable monotone form.
    Implicit throughout; validated empirically on dice events, not guaranteed.
  • standard math Variational inference standard assumptions (reparameterization, Gaussian posterior).
    Used for the VAE bounds in Equations (1)-(2) and Appendix A.
  • domain assumption True probabilities of dice events provide the correct ground truth.
    Used for evaluating accuracy; assumes the combinatorial computation is unambiguous.

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Cite this review

Pith. "Pith review of Recovering Event Probabilities from Large Language Model Embeddings via Axiomatic Constraints." pith.science (2026). https://pith.science/paper/LW3OCMMU

@misc{pith2026250507883,
  author       = {Pith},
  title        = {Pith review of: Recovering Event Probabilities from Large Language Model Embeddings via Axiomatic Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LW3OCMMU}},
  note         = {Machine review of arXiv:2505.07883}
}
read the original abstract

Rational decision-making under uncertainty requires coherent degrees of belief in events. However, event probabilities generated by Large Language Models (LLMs) have been shown to exhibit incoherence, violating the axioms of probability theory. This raises the question of whether coherent event probabilities can be recovered from the embeddings used by the models. If so, those derived probabilities could be used as more accurate estimates in events involving uncertainty. To explore this question, we propose enforcing axiomatic constraints, such as the additive rule of probability theory, in the latent space learned by an extended variational autoencoder (VAE) applied to LLM embeddings. This approach enables event probabilities to naturally emerge in the latent space as the VAE learns to both reconstruct the original embeddings and predict the embeddings of semantically related events. We evaluate our method on complementary events (i.e., event A and its complement, event not-A), where the true probabilities of the two events must sum to 1. Experiment results on open-weight language models demonstrate that probabilities recovered from embeddings exhibit greater coherence than those directly reported by the corresponding models and align closely with the true probabilities.

Figures

Figures reproduced from arXiv: 2505.07883 by the authors.

Figure 1
Figure 1. A schematic illustration of the two-step VAE-based learning algorithm. The objective is to [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Comparison of event probability estimates elicited from Gemma-2-9b-instruct: ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Visualization of the mean values of the 10 latent variables. Each panel compares the mean [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Relationship between judged probabilities (horizontal axis) and recovered probabilities [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Step 2 ablated. Visualization of the mean values of the 10 latent variables. Each panel compares the mean of the latent variables for embeddings corresponding to event A (horizontal axis) with those for embeddings corresponding to event not-A (vertical axis). 14 [PITH…
Figure 6
Figure 6. Figure 6: Alternative linear probe mapping LLM embeddings to true event probabilities. The left [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.