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For discrete random variables $X,Y,Z$,The difference between the left- and right-hand sides is the nonnegative conditional mutual information $I(X;Z\mid Y)$.

The axioms for information entropy give the formula $H(X)=-{\sum }_{x}p(x)\log p(x)$ and hence the chain rule for information entropyBecause conditioning reduces entropy, $H(Y\mid X)\le H(Y)$, and thereforeThis is subadditivity of information entropy.

For the entropy submodularity rule, apply the chain rule for information entropy twice:The last quantity is conditional mutual information, whose nonnegativity again expresses that conditioning reduces entropy.

Solved by gpt-5.6-sol high.

Suppose $X\to Y\to Z$ is a Markov chain, so $I(X;Z\mid Y)=0$. The chain rule for mutual information givesand alsobecause conditional mutual information is nonnegative. Therefore $I(X;Z)\le I(X;Y)$. Similarly,while $I(X,Y;Z)=I(X;Z)+I(Y;Z\mid X)\ge I(X;Z)$, so $I(X;Z)\le I(Y;Z)$. These are the two data processing inequalities. In particular, applying any deterministic function or Markov kernel to either argument cannot increase mutual information.

Solved by gpt-5.6-sol high.