Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 224 / 2 / c / Solution

Writing $Q_j$ for the $j$th marginal and using $P={\prod }_j{P}_j$,The analogous formula for $D(Q^{(i)}\Vert P^{(i)})$ omits coordinate $i$. ConsequentlyPart b makes ${\sum }_{i}H(Q^{(i)})\ge (n-1)H(Q)$, so the last display is at least $D(Q\Vert P)$.