A Lower Bound on Arbitrary $f$--Divergences in Terms of the Total Variation

TL;DR

This paper establishes a fundamental lower bound for all f-divergences based on the total variation, highlighting the central role of total variation in measuring differences between probability measures.

Contribution

It introduces a general lower bound for any f-divergence in terms of the total variation, under certain regularity conditions, which may be a novel insight.

Findings

Every f-divergence is bounded below by a monotonic function of total variation.

The bounding function is shown to be monotonic under regularity conditions.

The proof is straightforward, suggesting the result might be known in literature.

Abstract

An important tool to quantify the likeness of two probability measures are f-divergences, which have seen widespread application in statistics and information theory. An example is the total variation, which plays an exceptional role among the f-divergences. It is shown that every f-divergence is bounded from below by a monotonous function of the total variation. Under appropriate regularity conditions, this function is shown to be monotonous. Remark: The proof of the main proposition is relatively easy, whence it is highly likely that the result is known. The author would be very grateful for any information regarding references or related work.