Don't Pass$\mathtt{@}k$: A Bayesian Framework for Large Language Model Evaluation

• URL: http://arxiv.org/abs/2510.04265v1
• Date: Sun, 05 Oct 2025 16:14:03 GMT
• Title: Don't Pass$\mathtt{@}k$: A Bayesian Framework for Large Language Model Evaluation
• Abstract summary: Pass$@k$ is widely used to report performance for LLM reasoning, but it often yields unstable, misleading rankings.<n>We present a principled Bayesian evaluation framework that replaces Pass$@k$ with posterior estimates of a model's underlying success probability and credible intervals.
• Score: 4.082208996639461
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Pass$@k$ is widely used to report performance for LLM reasoning, but it often yields unstable, misleading rankings, especially when the number of trials (samples) is limited and compute is constrained. We present a principled Bayesian evaluation framework that replaces Pass$@k$ and average accuracy over $N$ trials (avg$@N$) with posterior estimates of a model's underlying success probability and credible intervals, yielding stable rankings and a transparent decision rule for differences. Evaluation outcomes are modeled as categorical (not just 0/1) with a Dirichlet prior, giving closed-form expressions for the posterior mean and uncertainty of any weighted rubric and enabling the use of prior evidence when appropriate. Theoretically, under a uniform prior, the Bayesian posterior mean is order-equivalent to average accuracy (Pass$@1$), explaining its empirical robustness while adding principled uncertainty. Empirically, in simulations with known ground-truth success rates and on AIME'24/'25, HMMT'25, and BrUMO'25, the Bayesian/avg procedure achieves faster convergence and greater rank stability than Pass$@k$ and recent variants, enabling reliable comparisons at far smaller sample counts. The framework clarifies when observed gaps are statistically meaningful (non-overlapping credible intervals) versus noise, and it naturally extends to graded, rubric-based evaluations. Together, these results recommend replacing Pass$@k$ for LLM evaluation and ranking with a posterior-based, compute-efficient protocol that unifies binary and non-binary evaluation while making uncertainty explicit. Code is available at https://mohsenhariri.github.io/bayes-kit

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