A note on the $f$-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters

TL;DR

This paper extends classical results on $f$-divergences between multivariate Gaussian distributions to broader location and scale families, providing practical tools for divergence comparison via Mahalanobis distances even when closed-form expressions are unavailable.

Contribution

It generalizes Ali and Silvey's results to multivariate location-scale families with prescribed parameters, enabling divergence comparisons through Mahalanobis distances.

Findings

$f$-divergences between isotropic Gaussians depend on Mahalanobis distance.

Conditions identified for generalizing divergence-distance relationship.

Extension of divergence results to multivariate scale families.

Abstract

We first extend the result of Ali and Silvey [Journal of the Royal Statistical Society: Series B, 28.1 (1966), 131-142] who first reported that any $f$-divergence between two isotropic multivariate Gaussian distributions amounts to a corresponding strictly increasing scalar function of their corresponding Mahalanobis distance. We report sufficient conditions on the standard probability density function generating a multivariate location family and the function generator $f$ in order to generalize this result. This property is useful in practice as it allows to compare exactly $f$-divergences between densities of these location families via their corresponding Mahalanobis distances, even when the $f$-divergences are not available in closed-form as it is the case, for example, for the Jensen-Shannon divergence or the total variation distance between densities of a normal location family. Second, we consider $f$-divergences between densities of multivariate scale families: We recall Ali and Silvey 's result that for normal scale families we get matrix spectral divergences, and we extend this result to densities of a scale family.