Minimization of divergences on sets of signed measures

TL;DR

This paper studies the problem of minimizing phi-divergences between a probability measure and subsets of signed measures, focusing on existence, characterization, and duality under linear constraints.

Contribution

It provides new results on the existence, characterization, and duality of phi-divergence projections onto sets of signed measures with linear constraints.

Findings

Established conditions for the existence of phi-projections.

Characterized the form of phi-projections under various constraints.

Analyzed duality and dual attainment in the context of signed measures.

Abstract

We consider the minimization problem of $\phi$-divergences between a given probability measure $P$ and subsets $\Omega$ of the vector space $\mathcal{M}_\mathcal{F}$ of all signed finite measures which integrate a given class $\mathcal{F}$ of bounded or unbounded measurable functions. The vector space $\mathcal{M}_\mathcal{F}$ is endowed with the weak topology induced by the class $\mathcal{F}\cup \mathcal{B}_b$ where $\mathcal{B}_b$ is the class of all bounded measurable functions. We treat the problems of existence and characterization of the $\phi$-projections of $P$ on $\Omega$. We consider also the dual equality and the dual attainment problems when $\Omega$ is defined by linear constraints.