Pinsker's inequality for adapted total variation

TL;DR

This paper extends Pinsker's inequality to the adapted total variation distance for stochastic process laws, showing it is bounded by a factor involving the square root of the process length and the relative entropy.

Contribution

It introduces a Pinsker-type inequality for the adapted total variation distance, applicable to laws of stochastic processes, with a bound involving process length and relative entropy.

Findings

Adapted total variation satisfies a Pinsker-type inequality.

The bound involves the square root of process length and relative entropy.

This extends classical inequalities to stochastic process laws.

Abstract

Pinsker's classical inequality asserts that the total variation $TV(\mu, \nu)$ between two probability measures is bounded by $\sqrt{ 2H(\mu|\nu)}$ 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 $\mu, \nu$ 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 $$ ATV(\mu, \nu)\leq \sqrt{n} \sqrt{2 H(\mu|\nu)}.$$