Metrization of powers of the Jensen-Shannon divergence

TL;DR

This paper proves that fractional powers of the Jensen-Shannon divergence can be metrized, confirming a conjecture and extending the approach to fractional powers of f-divergences between Cauchy distributions, with implications for statistical analysis.

Contribution

It provides a proof that fractional powers of Jensen-Shannon divergence are metrics and extends the method to Cauchy distributions, solving an open problem.

Findings

Fractional powers of Jensen-Shannon divergence are metrics.

Method applies to fractional powers of f-divergences between Cauchy distributions.

Confirms a conjecture from previous research.

Abstract

Metrization of statistical divergences is valuable in both theoretical and practical aspects. One approach to obtaining metrics associated with divergences is to consider their fractional powers. Motivated by this idea, Os\'an, Bussandri, and Lamberti (2018) studied the metrization of fractional powers of the Jensen-Shannon divergence between multinomial distributions and posed an open problem. In this short note, we provide an affirmative answer to their conjecture. Moreover, our method is also applicable to fractional powers of $f$-divergences between Cauchy distributions.