E-Scores for (In)Correctness Assessment of Generative Model Outputs (2510.25770v1)

Abstract: While generative models, especially LLMs, are ubiquitous in today's world, principled mechanisms to assess their (in)correctness are limited. Using the conformal prediction framework, previous works construct sets of LLM responses where the probability of including an incorrect response, or error, is capped at a desired user-defined tolerance level. However, since these methods are based on p-values, they are susceptible to p-hacking, i.e., choosing the tolerance level post-hoc can invalidate the guarantees. We therefore leverage e-values to complement generative model outputs with e-scores as a measure of incorrectness. In addition to achieving the same statistical guarantees as before, e-scores provide users flexibility in adaptively choosing tolerance levels after observing the e-scores themselves, by upper bounding a post-hoc notion of error called size distortion. We experimentally demonstrate their efficacy in assessing LLM outputs for different correctness types: mathematical factuality and property constraints satisfaction.

• The paper introduces e-scores as a novel method for post-hoc error control in generative models, addressing limitations of conventional p-scores.
• It details a construction based on reciprocals of e-values, providing statistical guarantees to bound size distortion under adaptive alpha selection.
• Experimental results demonstrate that e-scores effectively manage error rates across tests like mathematical factuality and property constraint satisfaction.

E-Scores for (In)Correctness Assessment of Generative Model Outputs

Introduction and Motivation

The assessment of correctness in generative model outputs, particularly those from LLMs, remains a critical challenge due to the prevalence of hallucinations and factual errors. Existing approaches, notably those based on conformal prediction, utilize p-values to construct response sets with controlled error rates. However, these methods are fundamentally limited by their susceptibility to p-hacking: the statistical guarantees are only valid if the tolerance level $\alpha$ is chosen independently of the observed data. In practice, users often wish to adapt $\alpha$ post-hoc, invalidating these guarantees.

This paper introduces e-scores, derived from e-values, as robust measures of incorrectness for generative model outputs. E-scores provide statistical guarantees for a post-hoc notion of error termed size distortion, enabling users to flexibly select tolerance levels after observing the scores. Theoretical analysis demonstrates that e-scores maintain valid guarantees under arbitrary post-hoc $\alpha$ strategies, and empirical results confirm their efficacy across diverse correctness criteria, including mathematical factuality and property constraint satisfaction.

Problem Formulation and Statistical Guarantees

The framework considers a generative model $\pi$ producing responses $\mathbf{y}$ to prompts $x \in \mathcal{X}$, with each response potentially decomposed into sub-responses (e.g., steps in chain-of-thought reasoning). Correctness is defined with respect to an oracle $o(x, \mathbf{y}) \in \{0, 1\}$, which is unknown at test time and must be estimated.

The central statistical desideratum is to upper bound the probability of including an incorrect response in a filtered set $\mathds{S}_\alpha$ by the tolerance level $\alpha$. For post-hoc $\alpha$, the guarantee is generalized to size distortion:

$E\left[\frac{1\{\exists\, \mathbf{Y} \in \mathds{S}_\alpha: o(X, \mathbf{Y}) = 0\}}{\alpha}\right] \leq 1$

where $\alpha$ may be chosen adaptively after observing the scored responses.

E-Score Construction and Theoretical Properties

E-scores are constructed as reciprocals of e-values, leveraging their property that $E[R] \leq 1$ for any e-variable $R$. For a test response $\mathbf{y}^{n+1}$, the e-score is defined as:

$s_{\text{e-score}}(x^{n+1}, \mathbf{y}^{n+1}) = \frac{n+1 \cdot f(x^{n+1}, \mathbf{y}^{n+1})}{f(x^{n+1}, \mathbf{y}^{n+1}) + \sum_{i=1}^n f^*(x^i, \mathds{O}(x^i, g_\pi(x^i)))}$

where $f$ is a non-negative function approximating the oracle, and $f^*$ is the maximal value over incorrect calibration responses.

Multiple choices for $f$ are considered, corresponding to different transformations of the oracle estimator $\hat{o}$, and combined via admissible averaging of e-values. Theoretical analysis (see Theorem 1) establishes that e-scores upper bound size distortion by 1 under exchangeable calibration and test data, even in the worst-case post-hoc $\alpha$ selection.

Practical Implementation

The e-score computation requires:

• A generative model $\pi$ for response generation.
• Calibration data with labeled correctness.
• An oracle estimator $\hat{o}$, typically a reward model or classifier trained independently.
• Efficient computation: e-scores require only a linear-time pass over calibration data, with constant memory per test response.

Algorithmically, for each test response, compute $f$ using $\hat{o}$, aggregate $f^*$ over calibration data, and apply the e-score formula. For multi-step responses, $\hat{o}$ can be applied at the sub-response level and aggregated multiplicatively.

Experimental Evaluation

Mathematical Factuality

Experiments on ProcessBench, a benchmark for mathematical reasoning, demonstrate that e-scores reliably control size distortion under both max-constrained adaptive $\alpha$ and fractional inclusion strategies. E-scores consistently yield mean error rates below or equal to the mean tolerance $\alpha$, in contrast to p-scores which often exceed $\alpha$.

Figure 1: Max-constrained adaptive $\pmb{\alpha}$ strategies for mathematical factuality, showing e-scores (orange) maintain size distortion below 1 across tolerance levels.

Precision-recall analysis indicates that e-scores are more conservative, prioritizing precision over recall, especially when $\alpha$ is restricted to $[0,1]$. The choice of oracle estimator (e.g., QwenPRM vs. MathShepherdPRM) significantly impacts precision and recall, highlighting the importance of accurate reward models.

Property Constraints Satisfaction

On UltraFeedback, e-scores are evaluated for instruction-following/helpfulness and truthfulness/honesty criteria. Again, e-scores maintain the size distortion guarantee, while p-scores fail under post-hoc $\alpha$ selection.

Figure 2: Instruction-following and helpfulness property constraints, with e-scores (orange) controlling size distortion and error rates under adaptive $\alpha$.

Precision-recall curves show that e-scores overlap with p-scores when recall is unconstrained, but become more conservative under strict $\alpha$ constraints.

Theoretical Implications and Extensions

The e-score framework generalizes previous conformal prediction approaches by supporting arbitrary post-hoc $\alpha$ strategies and larger response sets, including all permutations of sub-responses. This enables broader applicability, such as partial correctness assessment and flexible response filtering.

The theoretical results rely on exchangeability of calibration and test data, a standard assumption in conformal prediction. The framework is agnostic to the choice of generative model and oracle estimator, provided the latter is trained independently.

Implementation Considerations

• Computational Efficiency: E-scores are more efficient than p-scores, requiring only a single pass over calibration data.
• Oracle Estimator Quality: The accuracy of $\hat{o}$ directly affects the informativeness of e-scores; domain-specific reward models are recommended.
• Response Set Design: The framework supports flexible response set definitions, including partial and permuted responses.
• Deployment: E-scores can be integrated into LLM output pipelines for real-time correctness assessment and filtering.

Limitations and Future Directions

While e-scores provide robust post-hoc guarantees, their conservativeness may reduce recall in some settings. The choice and calibration of oracle estimators remain critical. Future work may explore alternative post-hoc error notions beyond size distortion, as well as improved oracle training strategies for specific domains.

Conclusion

E-scores offer a principled, efficient, and flexible mechanism for assessing the (in)correctness of generative model outputs under post-hoc tolerance selection. Theoretical and empirical results demonstrate that e-scores reliably control size distortion, outperforming p-score-based methods in adaptive scenarios. The framework generalizes to arbitrary generative models and response sets, with practical implications for safe and reliable deployment of LLMs in critical applications.