R eliable E val: A Recipe for Stochastic LLM Evaluation via Method of Moments (2025)

Abstract: LLMs are highly sensitive to prompt phrasing, yet standard benchmarks typically report performance using a single prompt, raising concerns about the reliability of such evaluations. In this work, we argue for a stochastic method of moments evaluation over the space of meaning-preserving prompt perturbations. We introduce a formal definition of *reliable evaluation* that accounts for prompt sensitivity, and suggest ReliableEval - a method for estimating the number of prompt resamplings needed to obtain meaningful results. Using our framework, we stochastically evaluate five frontier LLMs and find that even top-performing models like GPT-4o and Claude-3.7-Sonnet exhibit substantial prompt sensitivity. Our approach is model-, task-, and metric-agnostic, offering a recipe for meaningful and robust LLM evaluation.

## introduction a host of recent work has noticed that llms are highly sensitive to seemingly arbitrary prompt perturbations, throwing into question many of the results reported on popular benchmarks. these perturbations span various dimensions: semantically-equivalent paraphrases of the task instructions @xcite , changes in delimiters or whitespace @xcite , the order of in-context few-shot examples @xcite , among many others @xcite @xcite . while these works observed that llms are highly sensitive to prompt perturbations, to the best of our knowledge there is currently no prescriptive recipe for conducting meaningful evaluation which takes this sensitivity into account. evidently, many recent evaluation efforts resort to reporting llm performance against a single arbitrary prompt, while often acknowledging that this practice is flawed @xcite , highlighting the need for new evaluation practices. in this work, we argue that the evaluation of such sensitive llms requires stochastic evaluation over the spectrum of perturbations via a method of moments analysis (expected value, variance, etc.). to estimate moments over the combinatorially large perturbation sample space, we define the notion of reliable evaluation, which bounds the probability that a sample of prompt perturbations is representative of the entire sample space. further, we formulate reliableeval -a simple recipe for estimating the number of samples needed to achieve reliable evaluation per dataset. using our recipe, we perform stochastic evaluation of five frontier models, as well as leading open-source models, on three popular benchmarks. our findings, shown in figure 1 , reveal the statistical differences between models, highlighting the need for stochastic evaluation. moreover, we show that the number of resamplings required to reliably estimate model performance varies depending on both the model and the dataset being evaluated. we hope that our recommendations will be adopted to achieve meaningful and reliable reporting of llm performance. ## stochastic evaluation of llms: desiderata and approximation here we propose a set of desired metrics for llm evaluation in light of their observed sensitivity ( §2.1). since computing these metrics directly is infeasible, we also describe the desired statistical properties of a reliable approximation ( §2.2). in the following sections, we will operationalize these concepts ( §3), and use this approach to evaluate frontier llms ( §4). ## reliableeval: recipe for stochastic evaluation in this section, we present a practical recipe for conducting a reliable stochastic evaluation of llms. the recipe assumes a scenario aiming to evaluate a set of models m 1 , . . . , m k on a dataset d, while accounting for llms' sensitivity to meaningpreserving prompt perturbations. step 1: specify evaluation parameters ϵ and δ. set the acceptable deviation ϵ between the empirical value of the i-th moment over a sample s ′ ⊂ s d and the corresponding moment over the full distribution s d , as well as the confidence level δ with which this guarantee should hold, as defined in equation 4. in particular, we propose to set ϵ = 0.01 and δ = 0.1, i.e., that evaluation should be considered reliable if it deviates from true distribution by no more than 0.01 with probability of at least 0.9. this can critically examine claims of state of the art performance, which typically revolve around a difference of a few performance points between models @xcite . ## reliable stochastic evaluation of frontier models in this section, we present a reliable stochastic evaluation of five state-of-the-art llms, including both open-source and proprietary models, across three widely used benchmarks. ## related work most related to our work, polo et al. (2024) proposed a method for multi-prompt evaluation, hinging on a binary bernoulli distribution, limiting its applicability to text generation, and revolving around the selection of representative evaluation examples. in contrast, we find a minimal representative random subspace, are agnostic to the type of perturbations, and do not make any assumption about the scoring function. other works highlight the importance of multi-prompt evaluations , albeit without prescriptive guidelines @xcite @xcite . ## conclusion we propose to estimate model performance over prompt variations using moment analysis and show how to compute how many samples are needed for reliable results. our proposed method is designed to accommodate any computational budget, with an inherent trade-off between budget, error margin, and confidence. the practical question becomes: "given a specific compute budget, what is the most reliable evaluation achievable?" in our framework, the compute budget sets the maximum feasible n and constrains n * . if for a given error margin ϵ and confidence level δ the number of samples n * exceeds a given budget, it is still possible run the evaluation with @xmath0, accepting a larger margin of error or lower confidence as a result. thus, even with limited compute resources, our method provides guidance on how to maximize evaluation reliability within those constraints. finally, by evaluating frontier models across benchmarks, we find that sensitivity varies widely, underscoring the need for more robust evaluation practices.