Distances between nested densities and a measure of the impact of the prior in Bayesian statistics

Abstract

In this paper, we propose tight upper and lower bounds for the Wasserstein distance between any two univariate continuous distributions with probability densities $p_{1}$ and $p_{2}$ having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio $p_{1}/p_{2}$ as well as the Stein kernel $\tau_{1}$ of $p_{1}$. The method of proof relies on a new variant of Stein’s method which manipulates Stein operators.

We give several applications of these bounds. Our main application is in Bayesian statistics: we derive explicit data-driven bounds on the Wasserstein distance between the posterior distribution based on a given prior and the no-prior posterior based uniquely on the sampling distribution. This is the first finite sample result confirming the well-known fact that with well-identified parameters and large sample sizes, reasonable choices of prior distributions will have only minor effects on posterior inferences if the data are benign.