Bayesian nonparametrics, high dimensional techniques and simulation – Banhdits

Bayesian non parametric approaches have developped increasingly over the last 10 years, both in theory and applications. Indeed Bayes methods provide a useful paradigm that while retaining a fully model-based probabilistic framework, are flexible and adaptable. This project is concerned with development of Bayesian nonparametric methods according to three main (connected) themes: (1) Complex models (2) Asymptotics (3) Computational challenges. Axe 1: The first aim of our project is to devise and study theoretically powerful Bayesian "non parametric procedures " for complex models. In particular Bayesian approaches incorporate a sparseness-favoring structure which makes them specially relevant in high dimensional sparse models but there is no theoretical results on fully Bayes approaches. Extensions to models incorporating spatial information on the structure of the parameter, are of particular interest but the way the spatial coherence enables to improve on estimation is not understood yet. Other popular Bayesian non parametric models are based on the Dirichlet Process mixtures (as priors), which are typically used for density estimation and clustering and can be seen as infinite mixtures. Extensions to time-varying mixture models have recently been the focus of significant interest. In all the proposed models, the inference is challenging computationally difficult. Moreover, the dependence structure induced by such models is not clearly, neither have the frequentist properties of the posterior distributions been established. Dirichlet process mixtures (or variants) have been considered in particular in Poisson inverse problems. used for instance in Positron Emission Tomography for their coherence and their flexibility, however contrarywise to frequentist approches, the Bayesian literature on the subject is sparse and much needs to be done both theoreticaly and computationally. Another large class of statistical models is stochatic differential equations observed at discrete times. Several issues arise naturally in this context, both from a theoretical and computational angle, since the Bayesian literature on SDE is sparse. Axe 2: Even in simpler form of non parametric models characterizing the asymptotic behaviour the posterior distribution still remains an important challenge, despite the significant advances obtained since the years 2000, on consistency or on concentration rates in terms of distances that are comparable with the Kullback-Leibler divergence. Considering other distances is more complex. Also, the literature on non parametric Bernstein von Mises is very recent and much needs to be done. These results are useful for many reasons, they imply in particular that Bayesian and frequentist confidence regions are equivalent. An other aspect of inference is testing. Bayesian testing is mainly based on the Bayes factor. Consistency of the Bayes factor in nonparametric setups have been studied recently. However there exists no result on the determination of the separation rate of such tests. Axe 3: Finally computation of the procedure is a crucial issue of Bayesian non parametrics. Even though Monte Carlo methods such as MCMC (Markov chain Monte Carlo) have become standard in simple models, there are many open quesrions related to their application to nonparametric settings. Current solutions rely on very specific properties of the prior distribution and are thus difficult to generalise. One way to tackle the problem is to consider more recent methods, such as SMC (Sequential Monte Carlo) and Adaptive MCMC. Finally, from the opposite perspective, there has been little research on how to better assess the convergence and process the output of Monte Carlo methods, using non-parametric methods. Such ideas have appeared recently in the Approximate Bayesian Computation literature, where kernel density estimation has been used to improve on naive estimators, but much remains to be done in this direction.