An Iterative Procedure for Obtaining $I$-Projections onto the Intersection of Convex Sets

Abstract

A frequently occurring problem is to find a probability distribution lying within a set $\mathscr{E}$ which minimizes the $I$-divergence between it and a given distribution $R$. This is referred to as the $I$-projection of $R$ onto $\mathscr{E}$. Csiszar (1975) has shown that when $\mathscr{E} = \cap^t_1 \mathscr{E}_i$ is a finite intersection of closed, linear sets, a cyclic, iterative procedure which projects onto the individual $\mathscr{E}_i$ must converge to the desired $I$-projection on $\mathscr{E}$, provided the sample space is finite. Here we propose an iterative procedure, which requires only that the $\mathscr{E}_i$ be convex (and not necessarily linear), which under general conditions will converge to the desired $I$-projection of $R$ onto $\cap^t_1 \mathscr{E}_i$.