On the Impact of the Choice of the Prior in Bayesian Statistics

A key question in Bayesian analysis is the effect of the prior on the posterior, and how we can measure this effect. Will the posterior distributions derived with distinct priors become very similar if more and more data are gathered? It has been proved formally that, under certain regularity conditions, the impact of the prior is waning as the sample size increases. From a practical viewpoint it is more important to know what happens at finite sample size n. In this chapter, we shall explain how we tackle this crucial question from an innovative approach. To this end, we shall review some notions from probability theory such as the Wasserstein distance and the popular Stein’s method, and explain how we use these a priori unrelated concepts in order to measure the impact of priors. Examples will illustrate our findings, including conjugate priors and the Jeffreys prior.