Data processing inequality for four variable markov chain

Question

I came across this result in one of my lectures and I've been trying to prove it:

If $U \rightarrow X \rightarrow Y \rightarrow Z$, then $$ I \left( U; Z\right) \leq I \left(X;Y\right). $$

Can anyone provide a hint about how to approach the problem ?

Answer 1

If U, X, Y, Z forms a Markov chain, then the subchain $$U\rightarrow{Y}\rightarrow{Z}$$ is also a Markov chain. So to prove $$I(U;Z)\leq{I(X;Y)},$$ you need a middle variable $I(U;Y)$ , $$I(U;Z)\leq{I(U;Y)\leq{I(X;Y)}}.$$ These two inequalities can be proved using two different expansions of $I(U;Y,Z)$ and $I(Y;X,U)$ respectively.

Hope this could help!