Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 208 / 2 / b / Solution

Define the tilted probability measure $Q$ by $dQ=ZdP$; this is normalized because $\mathbb{E}Z=1$. ThenLet ${\mathbb{E}}_i$ average only coordinate $i$, keeping $X^{(i)}$ fixed. The marginal density of $Q_{X^{(i)}}$ relative to $P_{X^{(i)}}$ is ${\mathbb{E}}_{i}Z$, and henceMoreover,Substituting Han's inequality for relative entropy and rearranging gives the tensorization of entropy