Some Applications of Bayesian Statistics

This paper is intended as an introduction to Bayesian statistics for mathematicians who have no or very little previous experience with the subject. We start with a rather philosophical presentation of central concepts, as the philosophical approach to statistics differs from the standard approach of frequentist statistics. We also define probability distributions in a non-standard way, mostly to illustrate how "integration constants" can often be disregarded in Bayesian statistics, simplifying computations. We continue with a quick presentation of some central computational methods, followed by three longer examples of Bayesian data analysis and modelling. The goal is to give an impression of the applicability of the general concepts, and hopefully spark ideas for new applications in the reader