Separation and probability of correct classification among two or more distributions

Abstract

A function on theK-fold product of a set in normed vector space will be called a separation measurement if, for any collection ofK points, the function is bounded below and above, respectively, by maximum and total distance between pairs of points in the collection. Separation measurements are relavent toK-sample hypothesis testing and also to discrimination amongK classes, and several examples are given. In particular, ordinaryL 1 distance between integrable functions can be generalized to a non-pairwise separation measurement for densitiesf 1,f 2,...,f K inL 1[μ]; and this separation is a linear transform of the optimal discriminant's probability of correct classification.