When is subjective objective enough? Frequentist analysis of Bayesian methods

In this thesis, we investigate the properties of Bayesian methods. In particular, we want to give frequentist guarantees for Bayesian methods. A Bayesian starts with specifying their apriori belief as a probability distribution, the prior distribution. The prior is their inherently subjective beliefs. After a Bayesian has specified their prior, they collect data and compute the posterior distribution. For a Bayesian, this posterior distribution encodes their new beliefs on the world. However, this prior was subjective. Thus the posterior is also subjective. So we can wonder, will this posterior distribution give a better representation of reality? Will it be more accurate? The posterior distribution quantifies a subjective belief of uncertainty. How reliable is this quantification of uncertainty?
These questions lie at the foundation of this thesis. They have been answered for certain classes of prior distributions. However, they have not been fully answered for all distributions in use. In this thesis, in the introduction, we explain the foundational statistical theory to study these questions. In particular, we show how to apply Schwartz theorem and the Bernstein-von Mises theorems to study posterior distributions. We then turn to novel research.....