Bayesian nonparametric inference - Einzelansicht

Important change to schedule:

• The lectures on Thursday will begin at 08:45 --- not at 08:15 --- and end at 09:45.
• Further information about the lectures and the seminar will be given in the course.

This is an advanced course that will study some parts of the theory of Bayesian nonparametric inference.

Bayesian nonparametric inference is an important and active research area within mathematical statistics. The main goal is to infer an unknown infinite-dimensional parameter, such as a function, using observations from statistical experiments. The main object of interest is the posterior probability measure, which describes the distribution of the unknown parameter.

The course will be organised around 4-hour lectures and a 2-hour seminar.

Interested participants are required to register on PULS, or if this is not possible, to e-mail the lecturer.

Subhashis Ghosal and Aad van der Vaart, "Fundamentals of Nonparametric Bayesian Inference", Cambridge University Press, 2017.

Slides of the lectures will be provided.

In order to benefit from this course, participants should know the following concepts and tools, or be willing to learn them on their own. These concepts and tools will be used in proofs in the lectures.

Measure-theory: measurable spaces, measures, supports of measures, absolute continuity with respect to a measure, measurable mappings, Borel sigma-algebra, monotone convergence theorem, dominated convergence theorem, Fubini-Tonelli theorem, Fatou's lemma, etc.

Measure-theoretic probability theory: probability measures, random variables, pushforward measure / image measures, conditional expectations, different types of convergence of random variables, limit superior and limit inferior of events, Borel-Cantelli lemmas, Markov's inequality, Markov kernels, etc.

Real analysis: series and sequences, limits, limit inferior and limit superior, Taylor expansions, metric spaces

Functional analysis: function spaces (e.g. Hölder, Sobolev), norms on function spaces.

Students who take the course for 9 ECTS or 9 LP are required to give a seminar presentation and receive a passing grade for their presentation in order to take the oral exam.

Students who take the course for 6 ECTS or 6 LP are not required to give a seminar presentation in order to take the oral exam.

The oral exam will be 30 minutes in duration.

The lectures of this course will focus on proofs and some analytical examples, most often from regression problems.

-Brief overview of some results from probability theory

-Priors on function spaces

-Posterior consistency

-Posterior contraction rates

This course will not cover the following:

-Dirichlet priors or priors on spaces of probability measures

-Numerical computations

The target audience for this course are participants who have the following:

-a strong foundation in measure-theoretic probability and analysis;

-a strong interest in theoretical mathematical statistics, in particular Bayesian nonparametrics; and

-a large amount of self-motivation and endurance.