Divergence and Deformed Exponential Family

Abstract

The Kullback--Leibler divergence together with exponential families establishes the foundation of information geometry and is widely generalized. Among the generalization, we focus on the $(h,\ddot{I}„)$-divergence and $(h,\ddot{I}„)$-exponential families. We present a sufficient condition for the $(h,\ddot{I}„)$-divergence to induce a Hessian structure on an $(h,\ddot{I}„)$-exponential family. We also define the $(h,\ddot{I}„)$-dependence of random variables and prove a kind of the law of large numbers.

37 pages. Comments welcome!