Bayesian Statistics in Infinite Dimensions: Targeting Priors by Mathematical Analysis
Novel methods for understanding key aspects that are essential
to the future of Bayesian inference for high- or infinite-dimensional
models and data.
By combining his expertise on empirical processes and likelihood theory with his recent work on posterior contraction he will foremost lay a mathematical foundation for the Bayesian solution to uncertainty quantification in high dimensions.
Decades of doubt that Bayesian methods can work for high-dimensional models or data have in the last decade been replaced by a belief that these methods are actually especially appropriate in this setting. They are thought to possess greater capacity for incorporating prior knowledge and to be better able to combine data from related measurements. His premise is that for high- or
infinite-dimensional models and data this belief is not well founded, and needs to be challenged and shaped by mathematical analysis.
His central focus is the accuracy of the posterior distribution as quantification of uncertainty. This is unclear and has hardly been
studied, notwithstanding that it is at the core of the Bayesian method. In fact the scarce available evidence on Bayesian credible sets in high dimensions (sets of prescribed posterior probability) casts doubt on their ability to capture a given truth. He shall discover how this depends strongly on the prior distribution, empirical or hierarchical Bayesian tuning, and posterior marginalizaton, and therewith generate guidelines for good practice.
Professor Van der Vaart studies these issues in novel statistical settings (sparsity and large scale inference, inverse problems, state space models, hierarchical modelling), and connect to the most recent, exciting developments in general statistics.