We 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-scale family and the generator $f$ in order to generalize this result. In that case, one can compare exactly $f$-divergences between densities of these location families via their Mahalanobis distances. In particular, this proves useful when the $f$-divergences are not available in closed-form as it is the case for example for the Jensen-Shannon divergence between multivariate isotropic Gaussian distributions. Furthermore, we show that the $f$-divergences between these multivariate location-scale families amount equivalently to $f$-divergences between corresponding univariate location-scale families. We present several applications of these results.
翻译:我们推广了Ali和Silvey[皇家统计学会杂志:Series B,28.1(1966),131-142]的结果,后者首先报告说,两个北高斯多变量之间的任何美元波动均相当于相应的Mahalanobis距离的严格增加的卡路里功能,我们报告的标准概率密度功能产生了多变量位置大小的家庭和发电机元美元,以便概括这一结果。在这种情况下,可以精确地比较这些位置家庭之间通过其马哈拉诺比距离的密度之间的美元差异。特别是,当美元差异在封闭形式上不可用时,这证明是有用的,因为例如多变量高斯分布的Jensen-Shannon差异。此外,我们表明,这些多变量大小家庭之间的美元波动幅度相当于相应的非亚拉尼比斯距离的美元差异。