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

The Markov property gives $I(X_1;X_4\mid X_3)=I(X_2;X_4\mid X_3)=0$. The chain rule therefore yieldsandConditional on $X_4$, the factorization of the chain still gives $X_1\to X_2\to X_3$. The conditional data-processing inequality thus givesRearranging proves