Pinsker's inequality for adapted total variation

Abstract

Pinsker's classical inequality asserts that the total variation $TV(\hat{I}¼,\hat{I}½)$ between two probability measures is bounded by $\sqrt{2H(\hat{I}¼|\hat{I}½)}$ where $H$ denotes the relative entropy (or Kullback-Leibler divergence). Considering the discrete metric, $TV$ can be seen as a Wasserstein distance and as such possesses an adapted variant $ATV$. Adapted Wasserstein distances have distinct advantages over their classical counterparts when $\hat{I}¼,\hat{I}½$ are the laws of stochastic processes $(X_k{)}_{k=1}^n,(Y_k{)}_{k=1}^n$ and exhibit numerous applications from stochastic control to machine learning. In this note we observe that the adapted total variation distance $ATV$ satisfies the Pinsker-type inequality $$ATV(\hat{I}¼,\hat{I}½)\le \sqrt{n}\sqrt{2H(\hat{I}¼|\hat{I}½)}.$$