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