Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 164 / 1 / i / Solution

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.