BACKUP

Modern statistics and machine learning problems often feature data sitting in an ambient space of high dimension. Yet, methods such as random forests or deep neural networks have recently enabled remarkable performance even in such complex settings. One main reason is that the data can often be explained through a low dimensional structure, hidden to the statistician. In such settings, Bayesian methods such as spike-and-slab variable selection priors, Bayesian additive regression trees (BART), Bayesian deep neural networks or deep Gaussian processes are routinely used by statisticians as well as in physics, astronomy and genomics applications. Among the reasons for the popularity of Bayesian algorithms, one can mention: their flexibility, in that it is relatively easy to model the unknown structure underlying the data through the prior distribution; the broad range of computational methods available, including variational approximations; their ability to quantify uncertainty through so-called credible sets. While there are many empirical successes, there is an important need for understanding and validation for such methods. From the mathematical perspective, one would like to be able to understand and demonstrate under which conditions such algorithms effectively work. The BACKUP project aims at providing theoretical backup for such modern statistical algorithms, around three research avenues. First, new results will be obtained for high-dimensional models and latent variable settings using Bayesian posterior distributions, tackling important recent questions of multiple testing and variable selection. Second, foundational results will be obtained for complex methods such as random forests and Bayesian deep neural networks, both for posteriors and their variational approximations. Third, we will address the fundamental question of uncertainty quantification, by deriving optimal efficient confidence sets from well-chosen Bayesian credible regions.