Definition. Exponential weight and partition function [ftip-008J]
Definition. Exponential weight and partition function [ftip-008J]
For a finite KL-alignment instance, define the exponential weight and partition function by
\[ a_r(y)=\exp \left (\frac {r(y)}{\beta }\right ), \qquad Z_r=\mathbb E_{y\sim p}[a_r(y)]. \]Every \(a_r(y)\) is finite and strictly positive. Since \(p\) is a probability law on a nonempty finite set, \(0<Z_r<\infty \).
This weight and normalizer are the fixed-query form of [paes2026theoretical, Equation (2.2)].