On Relations Between Tight Bounds for Symmetric $f$-Divergences and Binary Divergences

TL;DR

This paper establishes conditions under which binary divergence measures serve as tight lower bounds for symmetric divergence measures, with applications in nonequilibrium physics.

Contribution

It introduces a sufficient condition linking binary and symmetric divergences, ensuring tight bounds under specific variance conditions.

Findings

Binary divergence measures provide tight lower bounds for symmetric divergences.

The bounds are tight when probability measures share the same variance.

Application demonstrated in nonequilibrium physics contexts.

Abstract

Minimizing divergence measures under a constraint is an important problem. We derive a sufficient condition that binary divergence measures provide lower bounds for symmetric divergence measures under a given triangular discrimination or given means and variances. Assuming this sufficient condition, the former bounds are always tight, and the latter bounds are tight when two probability measures have the same variance. An application of these results for nonequilibrium physics is provided.