Abstract

Asymptotics plays a crucial role in statistics. The theory of asymptotic consistency of Bayesian nonparametric procedures has been developed by many authors, including Schwartz (1965), Barron, Schervish and Wasserman (1999), Ghosal, Ghosh and Ramamoorthi (1999), Ghosal, Ghosh and van der Vaart (2000), Shen and Wasserman (2001), Walker and Hjort (2001), Walker (2004), Ghosal and van der Vaart (2007) and Walker, Lijoi and Prunster (2007). This theory is mainly based on existence of uniformly exponentially consistent tests, computation of a metric entropy and measure of a prior concentration around the true value of parameter. However, both the test condition and the metric entropy condition depend on models but not on prior distributions. Because a posterior distribution depends on the complexity of the model only through its prior distribution, it is therefore natural to explore appropriate conditions which incorporate prior distributions. In this thesis we introduce the Hausdorff entropy and an integration condition, both of which incorporate prior distributions and moreover are weaker than the metric entropy condition and the test condition, respectively. Furthermore, we provide an improved method to measure the prior concentration. By means of these new quantities, we derive several types of general posterior consistency theorems and general posterior convergence rate theorems for i.i.d. and non-i.i.d. models, which lead to improvements in a number of currently known theorems and their applications. We also study rate adaptation for density estimation within the Bayesian framework and particularly obtain that the Bayesian procedure with hierarchical prior distributions for log spline densities and a finite number of models achieves the optimal minimax rate when the true density is Hölder-continuous. This result disconfirms a conjecture given by Ghosal, Lember and van der Vaart (2003). Finally, we find a new both necessary and sufficient condition on Bayesian exponential consistency for prior distributions with the Kullback-Leibler support property.