On the Chi square and higher-order Chi distances for approximating f-divergences

TL;DR

This paper derives closed-form formulas for Chi square and higher-order Chi distances within exponential families, providing analytic expressions for f-divergences using Taylor expansions, applicable to Poisson and Gaussian distributions.

Contribution

It introduces new closed-form formulas for Chi distances and an analytic approach for f-divergences based on extended Chi-type distances in exponential families.

Findings

Closed-form formulas for Chi distances in exponential families

Analytic expressions for f-divergences using Taylor expansions

Application to Poisson and Gaussian distributions

Abstract

We report closed-form formula for calculating the Chi square and higher-order Chi distances between statistical distributions belonging to the same exponential family with affine natural space, and instantiate those formula for the Poisson and isotropic Gaussian families. We then describe an analytic formula for the $f$-divergences based on Taylor expansions and relying on an extended class of Chi-type distances.