AI 中文总结
研究变分自由能分解的唯一性问题，通过假设一对\((G,\Delta)\)满足特定条件，证明了\(G = F\)且\(\Delta = D(\cdot\,\|\,\pi)\)，给出了能精确分解的泛函的特征，与相对熵公理化表征互补。
AI 中文摘要
对于具有参考测度\(p\)和正权重\(\ell\)的有限系统，变分自由能\(F(Q;p,\ell)=D(Q\,\|\,p)-E_Q[\log\ell]\)满足精确恒等式\(\log Z(p,\ell)=-F(Q;p,\ell)+D(Q\,\|\,\pi)\)，其中\(Z\)是配分函数，\(\pi\)是相关的吉布斯测度。对于均匀参考测度，这是平均场理论的吉布斯 - 博戈留波夫不等式；对于贝叶斯模型，这是变分推断的证据分解。由\(\alpha\) - 或雷尼散度构建的变分目标在\(\log Z\)上保留有用的界，但不具有这种形式的恒等式，这就引出了哪些泛函允许精确分解的问题。我们证明了以下特征。假设一对\((G,\Delta)\)满足\(\log Z=-G+\Delta\)，其中\(\Delta\)仅是\(Q\)和\(\pi\)的函数；假设\(G\)可加分离为一个依赖于参考测度的项和一个依赖于权重的项，且具有适度的正则性；假设\(\Delta\)是非负的，并且恰好在\(Q = \pi\)时消失。那么\(G = F\)且\(\Delta = D(\cdot\,\|\,\pi)\)。证明将假设简化为从正函数的乘法群到\((\mathbb{R},+)\)的同态，并利用相对熵在其最小值处的二阶消失来消除它。反例表明每个假设都是必要的。该结果表征的是分解而非散度，因此与肖尔 - 约翰逊、西斯扎尔和阿马里给出的相对熵的公理化表征互补，既未使用也未扩展它们。
英文摘要
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 $Ψ_p(u)=\log\sum_x p(x)e^{u(x)}$ and $π_{p,u}(x)=p(x)e^{u(x)-Ψ_p(u)}$. The variational free energy $F(Q;p,u)=D(Q\|p)-\mathbb{E}_Q[u]$ satisfies the exact identity $Ψ_p(u)=-F(Q;p,u)+D(Q\|π_{p,u})$, 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 $Ψ_p(u)+G(Q;p,u)$ is nonnegative and vanishes whenever $Q=π_{p,u}$. Then $G=F$ and the gap equals $D(Q\|π_{p,u})$; $A$ 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 $α$-divergence distinct from Kullback--Leibler and no Rényi divergence of order other than one can occur as the residual.
Comments10 pages. Substantially revised and strengthened: removes the target-only and regularity assumptions, replaces the proof with a telescoping exponential-tilt argument, and expands the information-geometric framing and sharpness analysis. Prepared for submission to Information Geometry
