Monotonicity of Fisher-Rao Metric

Stochastic Map

A stochastic map is a linear map from the space of probability measures on some set to the space of probability measures on some set . Explicitly, we requirewhere is the point measure at .

Theorem

The relative entropy is monotone. i.e. for any stochastic map , we have

Proof By joint convexity of relative entropy, we have which is the desired result.