It is increasingly important to evaluate the characteristics of systems based on large language models (LLMs). Evaluations in this context often rely on a curated benchmark set of input prompts provided to the LLM, where the output for each prompt may be assigned a binary or ordinal score and the aggregation of scores across prompts is then used as a summary evaluation. In this paper, we develop a Bayesian approach for quantifying the uncertainty that arises in such evaluation metrics as a result of the stochasticity of the LLM-based systems; the same prompt may exhibit different outcomes on repeated runs. Our framework leads naturally to a sequential evaluation, in which we leverage the Bayesian model to preferentially select which prompts in the benchmark to use next, enabling more cost-effective LLM evaluations. We demonstrate this approach through four case studies: pairwise LLM preferences in interactive dialogue (MT-Bench), refusal rates on adversarial inputs (JailbreakBench), embodied agentic reasoning (ALFWorld), and agentic web-search Q&A (GAIA). Our results illustrate how a Bayesian approach can provide insightful uncertainty quantification about the behavior of blackbox LLM-based systems in a manner that is not accessible via deterministic or single-run evaluations.

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