Closed-form solutions to some generalized variational inference problems

Abstract

The Donsker--Varadhan formula characterizes the ordinary Bayesian posterior as the solution of an unrestricted $\mathsf{KL}$-regularized variational problem. Generalized variational inference replaces this regularizer by other divergences, but the resulting measure-valued optimization problem is often studied only after restriction to a parametric variational family. This paper studies the unrestricted measure-level problem. Given a measurable space $(\mathcal{Z},\mathfrak{Z})$, a prior probability measure $P$, a measurable loss $\ell:\mathcal{Z}\to(-\infty,\infty]$, a regularization strength $\alpha>0$, and a divergence $\mathsf{D}(Q\Vert P)$, we seek probability measures in \[ \underset{Q\in\mathcal{P}(\mathcal{Z})}{\mathrm{arg\,min}}\left\{\int_{\mathcal{Z}} \ell\,\mathrm{d}Q+\alpha\mathsf{D}(Q\Vert P)\right\}. \] For $f$-divergence penalties we derive a scalar inverse-gradient density formula and a one-dimensional dual identity; the Kullback--Leibler, Cressie--Read, and squared-Hellinger problems are treated as examples. Reverse $f$-divergences and mixed forward/reverse Kullback--Leibler penalties follow from the same separable integral principle. For Bregman divergences between densities we obtain a density-space solution with a scalar mass multiplier, including least-squares, density-power, and Burg/Itakura--Saito examples. For R\'enyi penalties of order $r>1$ we derive a normalized truncated-power characterization and a threshold equation for every global optimizer. Finite model-weight formulas and simple conjugate Bayesian model illustrations show how these closed forms are realized in practice and differ from the traditional solutions.