Contributions to Bayesian nonparametrics

Résumé

This dissertation focuses on the frequentist properties of Bayesian procedures in a broad spectrum of infinite-dimensional statistical models via Bayesian nonparametric approaches. Three essays concern the asymptotic aspects of posterior distribution in various statistical models presented in the subsequent three chapters. In the context of multivariate density estimation discussed in Chapter 2, a Bernstein-Dirichlet prior is constructed in the space of multivariate densities on hypercube and the corresponding posterior contraction rate is obtained. We implement this model through a novel sampling algorithm based on a slice sampling scheme for the simulated and real data. In Chapter 3, we consider a Bayesian semiparametric approach for a linear regression model with conditional moment restrictions. The error variable follows a Gaussian distribution whose variance depends on the predictors. An adaptive Bayesian procedure is performed when the priors on the conditional standard deviation function are carefully constructed. Chapter 4 is devoted to the issue of posterior convergence rate for a broad range of priors. Motivated by the boundary support estimation problems where any constructed prior could not meet the usual criteria for large sample analysis of the posterior distribution, we develop a new yardstick that allows flexible prior selections to counteract these problems by the stronger model conditions and meanwhile the rate optimality property is maintained.