  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.
