Chain rule for relative entropy
= Chain rule for relative entropy
For <probability distribution>[probability distributions] $P$ and $Q$ on a product space, the <Kullback-Leibler divergence> decomposes into the divergence of a marginal and the expected divergence of the corresponding conditional distributions. Iteration gives
$$
D(Q\Vert P)=\sum_i\mathbb E_QD\bigl(Q_i(\mathord\cdot\mid X_{<i})\Vert P_i(\mathord\cdot\mid X_{<i})\bigr).
$$