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

A frequently occurring problem is to find a probability distribution lying within a set E which minimizes the I-divergence between it and a given distribution R. This is referred to as the I-projection of R onto E. Csiszar (1975) has shown that when E = ∩ᵗ₁ Eᵢ is a finite intersection of closed, linear sets, a cyclic, iterative procedure which projects onto the individual Eᵢ must converge to the desired I-projection on E, provided the sample space is finite. Here we propose an iterative procedure, which requires only that the Eᵢ be convex (and not necessarily linear), which under general conditions will converge to the desired I-projection of R onto ∩ᵗ₁ Eᵢ.