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 $\cap^t_1 \mathscr{E}_i$.
Publié le : 1985-08-14
Classification:
$I$-divergence,
$I$-projections,
convexity,
Kullback-Liebler information number,
cross-entropy,
iterative projections,
iterative proportional fitting procedure,
90C99,
49D99
@article{1176992918,
author = {Dykstra, Richard L.},
title = {An Iterative Procedure for Obtaining $I$-Projections onto the Intersection of Convex Sets},
journal = {Ann. Probab.},
volume = {13},
number = {4},
year = {1985},
pages = { 975-984},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992918}
}
Dykstra, Richard L. An Iterative Procedure for Obtaining $I$-Projections onto the Intersection of Convex Sets. Ann. Probab., Tome 13 (1985) no. 4, pp. 975-984. http://gdmltest.u-ga.fr/item/1176992918/