An iterative procedure for obtaining I-projections onto the intersection of convex sets. (English) Zbl 0571.60006

Dykstra, Richard L.

Ann. Probab. 13, 975-984 (1985).

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. I. Csiszar [ibid. 3, 146-158 (1975; Zbl 0318.60013)] has shown that when \(E=\cap^ t_ 1E_ i\) is a finite intersection of closed, linear sets, a cyclic, iterative procedure which projects onto the individual \(E_ i\) 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 \(E_ i\) be convex (and not necessarily linear), which under general conditions will converge to the desired I-projection of R onto \(\cap^ t_ i\) \(E_ i\).

MSC:

60A99	Foundations of probability theory
90C99	Mathematical programming
94A17	Measures of information, entropy
49M99	Numerical methods in optimal control

Keywords:

Kullback-Leibler information; I-divergence; iterative procedure; I- projection

Citations:

Zbl 0318.60013