Jul 14, 2017

Medallion Lecture preview: Judith Rousseau

Judith Rousseau is currently Professor at University Paris Dauphine. Her research interests range from theoretical aspects of Bayesian procedures, both parametric and nonparametric, to more methodological developments. From a theoretical perspective she is interested in the interface between Bayesian and frequentist approaches, looking at frequentist properties of Bayesian methods. From a more methodological perspective, she has worked on MCMC or related algorithms or on the elicitation of subjective priors. She is an associate editor of the Annals of Statistics, Bernoulli, ANZJS and Stat, and is currently the program secretary of IMS. She has also been active on various aspects of the ISBA society. She is an ISBA and an IMS fellow and received the Ethel Newbold prize in 2015.

On the semi-parametric Bernstein von Mises Theorem in some regular and non regular models.

Bayesian nonparametrics has become a major field in Bayesian statistics, and more generally in statistics, over the last couple of decades with applications in a large number of fields within biostatistics, physics, economics, social sciences, computational biology, computer vision and language processing. Bayesian approaches are based on both a sampling model about observations given a parameter and on a prior model on the parameter.

With the elaboration of modern complex and large dimensional models, the need to understand their theoretical properties becomes crucial, in particular to understand what are the underlying assumptions behind the prior model. One way to shed light on such assumptions is to study the frequentist properties of the Bayesian procedures.

Consider a statistical model associated to a set of observations Yn Y(n) ~ Pϑ, ϑ Θ where n denotes a measure of information of the data Yn. Generally speaking Θ can have a very complex structure, be high- or infinite-dimensional. In a Bayesian approach, then one must additionally consider a prior model on the parameter through a probability distribution on Θ, called the prior distribution.

In large-dimensional models the influence of the prior is strong and does not entirely vanish asymptotically, i.e. when the information in the sample increases. It is then of interest to understand the types of implicit assumptions which are made by the choice of a specific prior and also within a family of priors which are the hyperparameters whose influence does not disappear as the number of observations increases.

Among the (many) advantages of Bayesian approaches is the fact that the inference is based on a whole probability distribution on the unknown parameter ϑ, namely the posterior distribution which is the conditional probability distribution of the parameter given the observations. With such a flexible tool, one can derive not only point estimators but also various measures of uncertainty. A common way to derive such measures of uncertainty is to construct credible regions, which are regions of the parameter space which have large posterior probability. These regions obviously depend on the prior distributions and it is important to understand how they are impacted by the assumptions (not necessarily explicit) made by the choice of the prior model. A way to do so is to study the asymptotic frequentist properties of these regions. The Bernstein–von Mises Theorem is a powerful tool to conduct such studies.

In my lecture I will describe some of the recent advances that have been obtained in the study of the Bernstein–von Mises theorem in large- and infinite-dimensional models, concentrating mainly in cases where only a finite-dimensional parameter is of interest in an infinite-dimensional model. I will consider both regular and irregular models.

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