LLMs Judging LLMs: A Simplex Perspective

TL;DR

This paper introduces a geometric simplex perspective to analyze when LLM-based judging of other LLM outputs is valid, proposing Bayesian priors to model judge uncertainty and improve ranking robustness.

Contribution

It provides a novel geometric framework for understanding LLM judging, with theoretical conditions and Bayesian methods to account for judge quality uncertainty.

Findings

Rankings are more reliable with two-level scoring than multi-level.

Bayesian methods improve coverage rates over existing procedures.

Modeling judge uncertainty enhances robustness of LLM-based evaluations.

Abstract

Given the challenge of automatically evaluating free-form outputs from large language models (LLMs), an increasingly common solution is to use LLMs themselves as the judging mechanism, without any gold-standard scores. Implicitly, this practice accounts for only sampling variability (aleatoric uncertainty) and ignores uncertainty about judge quality (epistemic uncertainty). While this is justified if judges are perfectly accurate, it is unclear when such an approach is theoretically valid and practically robust. We study these questions for the task of ranking LLM candidates from a novel geometric perspective: for $M$-level scoring systems, both LLM judges and candidates can be represented as points on an $(M-1)$-dimensional probability simplex, where geometric concepts (e.g., triangle areas) correspond to key ranking concepts. This perspective yields intuitive theoretical conditions and visual proofs for when rankings are identifiable; for instance, we provide a formal basis for the ``folk wisdom'' that LLM judges are more effective for two-level scoring ($M=2$) than multi-level scoring ($M>2$). Leveraging the simplex, we design geometric Bayesian priors that encode epistemic uncertainty about judge quality and vary the priors to conduct sensitivity analyses. Experiments on LLM benchmarks show that rankings based solely on LLM judges are robust in many but not all datasets, underscoring both their widespread success and the need for caution. Our Bayesian method achieves substantially higher coverage rates than existing procedures, highlighting the importance of modeling epistemic uncertainty.