My main interest lies in modelling with Bayesian methods. The scope of such modelling is vast, and quite a number of issues need to be addressed. The nonparametric Bayesian viewpoint provides a great perspective on how to model data and how to transport time-honored classical modelling strategies to the Bayesian world. As an example, classical statisticians have long divided terms in models into fixed effects and random effects. Early Bayesian efforts drew distinctions between the sorts of prior distributions recommended for these two types of effects through use of improper prior distributions for fixed effects and proper, normal prior distributions for random effects. Now, Bayesians generally like to have full support for their prior distributions. Standard priors (whether proper or improper) give full support for fixed effects. But for random effects, where the effects are modelled as draws from some arbitrary distribution, parametric priors do not have full support. A nonparametric Bayesian prior distribution is needed, and we now have the tools to work with this type of prior. The results of an analysis with a prior having full support can be quite a bit better than those of an analysis with a prior having small support (read a parametric analysis).
The paper with Chris Bush lays out the fixed effect/random effect argument and implements it in an analysis of the most common model involving fixed and random effects--the randomized block design. Other papers in this first batch describe further nonparametric Bayesian modelling strategies and lay out computational methods. Many of the computational methods developed in these papers, such as collapsing the state space of a Markov chain for Markov chain Monte Carlo methods, diagnosing poor convergence of a chain and then adding a step to improve performance, or deliberately introducing non-identifiability into a model to improve performance of a simulation are just as useful for parametric analyses as for nonparametric analyses.
Of the next two papers, the first provides a detailed example of prior elicitation and makes a case for considering asymptotics when eliciting the prior. The second describes a tool for Bayesian data analysis.
This paper shows that the idea of Bayesian inference as a compromise between the prior distribution and the likelihood carries with it strong implications for the form of the prior distribution. In fact, a compromise characterizes conjugate prior distributions!