When Can You Debias an LLM Judge? Identifiability Limits, a Test, and Designs for Top-k Ranking

Jian Xu, Delu Zeng, John Paisley, Qibin Zhao

Large language models (LLMs) are increasingly used as cheap, scalable judges that compare candidate outputs pairwise. Because such judges prefer verbose or well-formatted answers, the natural fix is to add bias covariates to a Bradley--Terry model and estimate the bias away. We show this cannot work as advertised: the quality/bias split is \emph{not identified} by pairwise comparisons, and the failure is exact -- across $48$ real judge-pools the profile likelihood over the coefficient is flat to $\mathbf{0.0000}$ \textbf{nats}, and scaling the comparisons $26\times$ buys none. A ``debiased'' score is selected by the prior, not recovered from data. Our contribution is accordingly not a better estimator but a characterization of \emph{when prior-based correction is justified}, plus designs that supply the missing information when it is not. The assumption the prior encodes -- quality is a priori uncorrelated with the covariate -- pays only while $\mathrm{corr}(θ,x)$ stays below a crossing point (configuration-dependent, $0.22$--$0.60$), which is what makes the same model help on LLMBar and hurt on SummEval and Nectar. We give two escapes: a \textbf{trusted-anchor gate} that decides per (judge, covariate, task) (no false enables in $6{,}000$ decisions at $K\ge10$ anchors, a rate our sample bounds at $\le6\%$), and a \textbf{paired rendering design}. Across fifteen real LLM judges bias is heterogeneous and capability-dependent: correction improves \topk{} recall by $0.20$--$0.32$ on five biased-but-competent cheap judges and is a no-op on frontier ones (Spearman $ρ{=}{-}0.84$ between competence and gain over the $14$ competent judges, $p{<}10^{-3}$), concentrating the benefit where at-scale evaluation happens.