Probability theory, basic statistical concepts (estimation, testing, confidence sets), analysis.
Probability is assumed at the level of measure theory, as in the mastermath course measure-theoretic probability. Knowledge of asymptotic statistics, e.g. as in the mastermath course of that title, is recommended.

Aim of the course

Introduction to the theory and practice of Bayesian statistics.

Bayesian statistics is routinely used in various fields of science and applications, and perhaps is experiencing a revival in popularity. In the Bayesian approach one first specifies a so-called prior probability distribution on the parameter space, which represents the initial belief or expert knowledge about the problem. Next, after obtaining data, one updates this prior to the conditional distribution of the parameter given the data, the so-called posterior distribution.

The aim of this course is to give a rigorous, measure-theoretic, introduction to Bayesian statistical procedures and investigate its performance in the usual framework, in which it is assumed that the data are generated according to a given parameter. We shall be concerned with the question whether the posterior distribution is able to reconstruct this parameter, for instance if the amount of data would increase indefinitely, and whether it is reliable for uncertainty quantification. We shall study specific examples of prior distributions (both parametric conjugate and objective ones, and nonparametric ones, such as the Dirichlet process and Gaussian processes). We shall also address the computation of posterior distributions, in particular methods based on simulating a Markov chain whose distribution approximates the posterior distribution (MCMC methods).

A. van der Vaart (UL)
B. Szabó (UL)