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تسجيل مستخدم جديدProjection Theorems for the Renyi Divergence on $alpha$-Convex Sets
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This paper studies forward and reverse projections for the R{e}nyi divergence of order $alpha in (0, infty)$ on $alpha$-convex sets. The forward projection on such a set is motivated by some works of Tsallis {em et al.} in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoes proved a Pythagorean inequality for R{e}nyi divergences on $alpha$-convex sets under the assumption that the forward projection exists. Continuing this study, a sufficient condition for the existence of forward projection is proved for probability measures on a general alphabet. For $alpha in (1, infty)$, the proof relies on a new Apollonius theorem for the Hellinger divergence, and for $alpha in (0,1)$, the proof relies on the Banach-Alaoglu theorem from functional analysis. Further projection results are then obtained in the finite alphabet setting. These include a projection theorem on a specific $alpha$-convex set, which is termed an {em $alpha$-linear family}, generalizing a result by Csiszar for $alpha eq 1$. The solution to this problem yields a parametric family of probability measures which turns out to be an extension of the exponential family, and it is termed an {em $alpha$-exponential family}. An orthogonality relationship between the $alpha$-exponential and $alpha$-linear families is established, and it is used to turn the reverse projection on an $alpha$-exponential family into a forward projection on a $alpha$-linear family. This paper also proves a convergence result of an iterative procedure used to calculate the forward projection on an intersection of a finite number of $alpha$-linear families.
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