Meaning-Dependent Judge Sensitivity Changes the Minimax Rate of Semantic Entropy

Semantic entropy, the entropy of the meanings among a language model's sampled answers, is estimated from noisy pairwise judgments. Its minimax theory assumes a judge that recognizes every meaning equally well. We show that a meaning-dependent sensitivity changes the minimax rate even when it is known exactly and the judge recognizes every meaning at least as well as a homogeneous one. Two answers with meaning \( c \) are declared equivalent with probability \( q+\Delta w_c \), two with different meanings with probability \( q \), where \( q \) is bounded away from 0 and 1 and the number \( K\ge3 \) of meanings is fixed. For almost every sensitivity profile \( w \), the entropy is uniformly consistently estimable exactly when \( N:=n^{2K-1}\Delta^{4K-4}\to\infty \): for \( \Delta=n^{-\beta} \) the threshold falls from \( \beta<K/(2K-2) \) for a homogeneous judge to \( \beta<(2K-1)/(4K-4) \), that of such a judge on \( 2K-1 \) meanings, and from 3/4 to 5/8 for three meanings. On every fixed polynomial window \( n\Delta^2\le n^{-\kappa} \) the minimax squared risk is of order \( \min\{1,N^{-1/(2K-1)}\} \). Star statistics computable in \( O(n^2K) \) time and a projection attain this rate through a weighted entropy modulus with optimal exponent \( 1/(2K-1) \). The lower bound uses a moment-preserving path near the interior point \( \theta\propto1/w \), along which the judge's channel moves with the distribution.