PULS
Foto: Matthias Friel
This is an advanced course on mathematical statistics that will focus on the theory of Bayesian nonparametric statistics. It is targeted for students who are strongly interested in proofs.
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.
In order to benefit from this course, participants should know the following, or be prepared to learn the material on their own.
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
Participants who take the course for 9 LP must give one seminar presentation of 90 minutes. The seminar topic will be arranged by the lecturer.
Participants who take the course for 6 LP are not required to give a seminar presentation.
All participants who take the course for credit must take a 30-minute oral examination. The examination will be based only on the material from the lectures.
The participant's grade on the exam will be their grade for the course.
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
If time permits, we will also cover additional advanced topics in Bayesian nonparametric inference, such as adaptive posterior contraction.
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 very strong foundation in measure-theoretic probability and analysis;
-a strong interest in proofs; and
-a large amount of self-motivation and endurance.
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