Refined Pinsker’s and Reverse Pinsker’s Inequalities for Probability Distributions of Different Dimensions

Abstract

We provide optimal lower and upper bounds for the augmented Kullback-Leibler divergence in terms of the augmented total variation distance between two probability measures defined on two Euclidean spaces having different dimensions. We call them refined Pinsker’s and reverse Pinsker’s inequalities, respectively.

Keywords

• Kullback-Leibler divergence
• total variation distance
• optimal bounds
• probability measures of different dimensions