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

The chain rule, now separating $X_i$ first, giveswhere the second term denotes the conditional divergence averaged over $Q_{X_i}$. Summing over $i$ yieldsRepeated use of the chain rule and convexity gives the tensorization lower bound for relative entropyTherefore the preceding sum is at least $(n-1)D(Q\Vert P)$, which is equivalent to the required inequality.