The Annals of Probability

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

Richard L. Dykstra

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 90C99: None of the above, but in this section
Secondary: 49D99

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


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.

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