Past exam of the information theory course of the University of Oxford / 2021 / 1 / a / i

The chain rule for conditional entropy givesIndeed, for every conditioning value $(x,z)$ of positive probability, each summand $-p(y\mid x,z){\log }_{2}p(y\mid x,z)$ is nonnegative. Its sum is zero exactly when the conditional probability mass function is concentrated at one value. Averaging over $(X,Z)$ therefore proves both the inequality and its equality criterion: equality holds exactly when $Y=f(X,Z)$ almost surely for some function $f$. Conditioning values of probability zero impose no restriction. This is a statement about discrete random variables; a differential entropy need not be nonnegative.