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In this paper we propose tight upper and lower bounds for the Wassersteindistance between any two {{univariate continuous distributions}} withprobability densities $p_1$ and $p_2$ having nested supports. These explicitbounds 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 proofrelies on a new variant of Stein's method which manipulates Stein operators.  We give several applications of these bounds. Our main application is inBayesian statistics : we derive explicit data-driven bounds on the Wassersteindistance between the posterior distribution based on a given prior and theno-prior posterior based uniquely on the sampling distribution. This is thefirst finite sample result confirming the well-known fact that withwell-identified parameters and large sample sizes, reasonable choices of priordistributions will have only minor effects on posterior inferences if the dataare benign.

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