Expected relative entropy decomposition ($\mathbb{E}D(Q\Vert r)=\mathbb{E}D(Q\Vert p)+D(p\Vert r)$)

Let $Q$ be a random probability mass function on a finite alphabet, let $p=\mathbb{E}Q$, and suppose a fixed reference $r$ is positive on the probability support of $p$. Then $Q$ vanishes almost surely wherever $p=0$, andTaking expected values yields the displayed decomposition. In particular it applies to an unbiased empirical distribution. It separates sampling fluctuation from reference mismatch: even perfect recovery of $p$ leaves the deterministic relative entropy $D(p\Vert r)$. A bound comparing different references must account for this term.