## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 3 (1985), 975-984.

### 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 $\cap^t_1 \mathscr{E}_i$.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 3 (1985), 975-984.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992918

**Digital Object Identifier**

doi:10.1214/aop/1176992918

**Mathematical Reviews number (MathSciNet)**

MR799432

**Zentralblatt MATH identifier**

0571.60006

**JSTOR**

links.jstor.org

**Subjects**

Primary: 90C99: None of the above, but in this section

Secondary: 49D99

**Keywords**

$I$-divergence $I$-projections convexity Kullback-Liebler information number cross-entropy iterative projections iterative proportional fitting procedure

#### Citation

Dykstra, Richard L. An Iterative Procedure for Obtaining $I$-Projections onto the Intersection of Convex Sets. Ann. Probab. 13 (1985), no. 3, 975--984. doi:10.1214/aop/1176992918. https://projecteuclid.org/euclid.aop/1176992918