论文标题
关于具有规定比例矩阵或位置参数的多元位置尺度家庭之间$ f $ divergences的注释
A note on the $f$-divergences between multivariate location-scale families with either prescribed scale matrices or location parameters
论文作者
论文摘要
我们首先扩展了Ali和Silvey的结果[皇家统计协会杂志：B系列，28.1（1966），131-142]，他们首先报道说，两个各向同性多元高斯分布之间的任何$ f $ divergence量相应地相应地增加了他们相应的Mahalananobis距离的标量。我们报告了产生多元位置族的标准概率密度函数和功能发生器$ f $的足够条件，以概括此结果。该属性在实践中很有用，因为它允许通过相应的玛哈拉诺比斯距离在这些位置家族之间的密度之间进行$ f $ diverence进行比较，即使$ f $ divergences无法以封闭形式可用，例如，对于Jensen-Shannon差异或正常位置家族密度之间的总变化距离，也是如此。其次，我们考虑了多元尺度家庭密度之间的$ f $ diverence：我们回想起阿里（Ali）和西尔维（Silvey）的结果，对于正常尺度的家庭，我们获得了矩阵频谱差异，我们将此结果扩展到了量表家族的密度。
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.
