Hotelling Lectures: Aad van der Vaart, Leiden University

Nonparametric Bayesian methods: frequentist analysis
Aad van der Vaart
Leiden University

A more detailed view of Bayesian methods to estimate functions or high-dimensional
parameter vectors, and discuss the validity (or not) of these methods from a
non-Bayesian point of view. For instance, we consider using a Gaussian process
as a prior for a regression function or (after exponentiation and normalisation) for a
density function. We characterise the rate at which the corresponding posterior distribution
can recover a true function as the noise level tends to zero or the number of observations tends to infinity,
and discuss how this rate can be improved by scaling the time axis, showing that an appropriate random
scaling leads to adaptive recovery over a scale of smoothness levels. Recovery means that the posterior
distribution concentrates most of its mass near the parameter that generates the data, for most
observations. It refers mostly to the location of the posterior distribution. A second use of the
posterior distribution is uncertainty quantification, and refers to the spread of the posterior distribution. In fact,
it is at the core of the Bayesian method to use the full posterior distribution as an indication
of remaining uncertainty. We discuss the general difficulties of uncertainty quantification in
nonparametric statistics, from which Bayesian methods of course also cannot escape. We argue that
these difficulties imply that the uncertainty quantification of adaptive Bayesian methods must
be misleading for certain true parameters, and present concrete examples.
We next show that for so-called self-similar parameters the uncertainty quantification is valid.

The first talk is a general introduction to these aspects of nonparametric Bayesian
statistics, focused mostly at curve estimation. In the second talk we also address
similar issues in the Bayesian recovery of regression parameters in sparse high-dimensional models.

Refreshments will be served prior to the lecture at 3:00PM in the 3rd floor lounge of Hanes Hall.