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August, 1985 An Iterative Procedure for Obtaining $I$-Projections onto the Intersection of Convex Sets
Richard L. Dykstra
Ann. Probab. 13(3): 975-984 (August, 1985). DOI: 10.1214/aop/1176992918


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


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Richard L. Dykstra. "An Iterative Procedure for Obtaining $I$-Projections onto the Intersection of Convex Sets." Ann. Probab. 13 (3) 975 - 984, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0571.60006
MathSciNet: MR799432
Digital Object Identifier: 10.1214/aop/1176992918

Primary: 90C99
Secondary: 49D99

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

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 3 • August, 1985
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