Sample complexity bounds for the Jensen-Shannon divergence

Abstract

The Jensen-Shannon divergence (JSD) is a symmetric and bounded measure of the dissimilarity of two probability distributions, which has become a standard tool in statistics, information theory, and machine learning. We complement the understanding of its mathematical properties by presenting an analysis of the amount of data that is needed to distinguish between two distributions, given the value of JSD between them. We find the number of independent and identically distributed samples that suffice for a classifier to determine which of two distributions generated observed data at a desired error rate, for two complementary classifiers: we show that for the log-likelihood-ratio classifier, a sample size that grows as the inverse JSD is sufficient, whereas for a majority-vote classifier assembled from independent single-sample decisions, the sufficient size grows as the squared inverse JSD. These distinct scalings offer operational readings of JSD values and their translation into distinguishability in different contexts.