A Uniqueness Theorem for Exact Separable Log-Partition Decompositions

Abstract

Let $\mathcal{X}$ be a finite set, let $p$ and $Q$ be strictly positive probability distributions on $\mathcal{X}$, and let $u\in\mathbb{R}^{\mathcal{X}}$ be a log-weight. Define $\Psi_p(u)=\log\sum_x p(x)e^{u(x)}$ and $\pi_{p,u}(x)=p(x)e^{u(x)-\Psi_p(u)}$. The variational free energy $F(Q;p,u)=D(Q\|p)-\mathbb{E}_Q[u]$ satisfies the exact identity $\Psi_p(u)=-F(Q;p,u)+D(Q\|\pi_{p,u})$, whose residual is the canonical divergence of the dually flat simplex. We prove a converse. Suppose $G(Q;p,u)=A(Q,p)+B(Q,u)$ is additively separable and its gap $\Psi_p(u)+G(Q;p,u)$ is nonnegative and vanishes whenever $Q=\pi_{p,u}$. Then $G=F$ and the gap equals $D(Q\|\pi_{p,u})$; $A$ and $B$ are unique up to a $Q$-dependent additive gauge. The gap is not assumed to be a divergence or to depend only on the target, and no continuity, measurability, differentiability, convexity, or membership in a prescribed divergence family is assumed. The proof bounds increments of a corrected accuracy term by centered log-moment-generating costs of order $k^{-2}$ per step. Telescoping $k$ equal exponential tilts gives an $O(k^{-1})$ bound and forces every increment to vanish. Thus target factorization and strictness are conclusions. Counterexamples show that separability, nonnegativity, and tightness are each necessary. Within this class, no $\alpha$-divergence distinct from Kullback--Leibler and no R\'enyi divergence of order other than one can occur as the residual.