Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 224 / 1 / a / Solution

If $X\to Y\to Z$ is a Markov chain, the two forms of the data processing inequality for mutual information areThe chain rule for mutual information and the Markov property $I(X;Z\mid Y)=0$ giveUsing the other order,because conditional mutual information is nonnegative. This proves the first inequality. Applying the same result to the reversed Markov chain $Z\to Y\to X$, which has the same conditional-independence statement, proves the second.