I am having a little trouble getting my head around the two "directions" of the Kullback-Leibler divergence:

Definition (Kullback-Leibler divergence)For discrete probability distributions $P$ and $Q$ defined on the same probability space, $\chi$, the Kullback-Leibler divergence from $Q$ to $P$ is defined to be $$D_{KL}(P||Q) := \sum_{x \in \chi}P(x)\log\bigg(\frac{P(x)}{Q(x)}\bigg). $$

Note that The KL-divergence is not really a true distance measure, since it is does not satisfy the Triangle Inequality and $D_{KL}(P||Q)$ does not in general equal $D_{KL}(Q||P)$. Hence the need to distinguish the KL-divergence from its dual, the so-called "reverse KL-divergence".

**Fact.** Both the KL-divergence and the reverse KL-divergence are examples of *f*-divergences.

**Fact.** The KL-divergence is an example of a Bregman divergence.

**Question:** Is the reverse Kullback-Leibler divergence also a Bregman divergence?

This is not obvious to me. Also, I have read conflicting information, with Amari (2009) arguing that the KL-divergence and its dual are are unique divergences belonging to the $f$-divergence and Bregman divergence classes, and others (e.g., Wang et al. 2019) saying that the reverse KL-divergence is no longer a Bregman. Which is correct?

Any pointers would be greatly appreciated.

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