Divergence and Deformed Exponential Family

TL;DR

This paper introduces a generalized divergence and exponential family framework, establishing conditions for geometric structures and proving a law of large numbers within this new setting.

Contribution

It presents a sufficient condition for the $(h, au)$-divergence to induce a Hessian structure and defines $(h, au)$-dependence, extending information geometry theory.

Findings

$(h, au)$-divergence induces Hessian structures under specific conditions

Defined $(h, au)$-dependence of random variables

Proved a law of large numbers for the $(h, au)$-framework

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.