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arXiv · 2204.10952

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

Frank Nielsen·Kazuki Okamura
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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.

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Frank Nielsen, Kazuki Okamura. 2022-05-31. A note on the $f$-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters. https://doi.org/10.1007/s11222-023-10373-6

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