On projections of the Rényi divergence on generalized convex sets

Motivated by a recent result by van Erven and Harremoës, we study a forward projection problem for the Rényi divergence on a particular α-convex set, termed αlinear family. 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 αexponential family. An orthogonality relationship between the αexponential and α-linear families is first established and is then used to transform the reverse projection on an α-exponential family into a forward projection on an α-linear family. The full paper version of this work is available on the arXiv at http://arxiv.org/abs/1512.02515. Index Terms – α-convex set, relative entropy, variational distance, forward and reverse projections, Rényi divergence, exponential and linear families.