On the Jensen-Shannon Divergence and Variational Distance

Abstract

Abstract-We study the distance measures between two probability distributions via two dierent distance metrics, a new metric induced from Jensen-Shannon Divergence[4] and the well known L1 metric. First we show that the bounds between these two distance metrics are tight for some particular distributions. Then we show that the L1 distance of a binomial distribution does not imply the entropy power inequality for the binomial family, proposed in [5]. Moreover, we show that, several important results and constructions in computational complexity under the L1 metric carry over to the new metric, such as Yao’s next-bit predictor [13], the existence of extractors [11], the leftover hash lemma[?] and the construction of expander graph based extractor. Finally we show that the useful parity lemma [12] in studying pseudo-randomness does not hold in the new metric.