Equality relationship between Total Variation and KL divergence

Question

I noticed that according to Pinsker's inequality, the following relationship can be got.

$D_{TV}(p||q) \le \sqrt{{1\above{1pt}2}D_{KL}(p||q)} $.

I'm curious whether, under some condition, the equality relationship between them can be got

$D_{TV}(p||q) = \sqrt{{1\above{1pt}2}D_{KL}(p||q)} $

Answer 1

For $ p ≠ q $ it is not generally possible. The reason is that these two distances measure different aspects of the dissimilarity between the distributions. In some specific cases, the equality might be achieved, but these cases are not general and would depend on particular properties of the probability distributions $p$ and $q$. In general, Pinsker's inequality only provides an upper bound on the total variation distance using the Kullback-Leibler divergence, and the relationship between the two is not generally an equality.