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 $(h,\tau)$-divergence and $(h,\tau)$-exponential families. We present a sufficient condition for the $(h,\tau)$-divergence to induce a Hessian structure on an $(h,\tau)$-exponential family. We also define the $(h,\tau)$-dependence of random variables and prove a kind of the law of large numbers.