Generalized Beta Divergence

TL;DR

This paper introduces a generalized form of beta divergence using a compact integral representation, simplifying property derivations and establishing its equivalence with statistical deviance.

Contribution

It extends beta divergence beyond classical forms with a new integral representation, linking it to statistical deviance and simplifying analysis.

Findings

Generalized beta divergence expressed as a compact integral.

Simplified derivations of scaling, translation, and expectation properties.

Established equivalence between beta divergence and statistical deviance.

Abstract

This paper generalizes beta divergence beyond its classical form associated with power variance functions of Tweedie models. Generalized form is represented by a compact definite integral as a function of variance function of the exponential dispersion model. This compact integral form simplifies derivations of many properties such as scaling, translation and expectation of the beta divergence. Further, we show that beta divergence and (half of) the statistical deviance are equivalent measures.