Bregman divergences based on optimal design criteria and simplicial measures of dispersion

TL;DR

This paper introduces a new class of divergences based on optimal design criteria and simplicial measures, connecting geometric dispersion with information-theoretic divergences, and empirically tests their ability to distinguish normal distributions.

Contribution

It develops a general framework for divergences using directional derivatives of functions, linking simplicial distances with optimal design criteria, and provides matrix formulas for normal distributions.

Findings

Sufficient conditions for divergence properties are established.

Matrix formulas for multivariate normal divergences are derived.

Empirical tests demonstrate effectiveness in distinguishing distributions.

Abstract

In previous work the authors defined the k-th order simplicial distance between probability distributions which arises naturally from a measure of dispersion based on the squared volume of random simplices of dimension k. This theory is embedded in the wider theory of divergences and distances between distributions which includes Kullback-Leibler, Jensen-Shannon, Jeffreys-Bregman divergence and Bhattacharyya distance. A general construction is given based on defining a directional derivative of a function $\phi$ from one distribution to the other whose concavity or strict concavity influences the properties of the resulting divergence. For the normal distribution these divergences can be expressed as matrix formula for the (multivariate) means and covariances. Optimal experimental design criteria contribute a range of functionals applied to non-negative, or positive definite, information matrices. Not all can distinguish normal distributions but sufficient conditions are given. The k-th order simplicial distance is revisited from this aspect and the results are used to test empirically the identity of means and covariances.