Don’t Pass@$k$: A Bayesian Framework for Large Language Model Evaluation

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, and BrUMO, 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.