The project will focus on research in the six areas of Robust Bayesian Analysis, Default Prior Bayesian Methodology, Conditional Inference in Decision Theory, Multivariate Estimation, Hierarchical Modelling, and Bayesian Computation. In the first two areas, special emphasis will be paid to problems involving testing and model selection. Bayesian approaches to these problems are severely underutilized, primarily because robust or default prior analyses in these areas have not been developed. In Conditional Inference, the primary focus will be the study of situations in which Bayesian answers have a valid conditional frequentist interpretation. Identification of such situations is not only valuable foundationally, but also indicates situations in which adopting a conditional frequentist approach can be highly advantageous. In Bayesian Computation, emphasis will be on use of Markov Chain and other simulation techniques for carrying out Bayesian integrations. Bayesian analysis is perhaps the most rapidly growing approach to statistical analysis, because of its ability to model and analyze even extremely complex situations, and because it readily allows combination of multiple information sources. The chief limitations of Bayesian analysis have been concern about sensitivity to modelling assumptions, and the difficulty in carrying out Bayesian computations in high dimensional problems. In this project we will attack the sensitivity problem by developing Bayesian models that are inherently "robust," and by creating powerful methods for selecting from among competing Bayesian models. On the computational side, we will extend new techniques based on simulation that have the promise of handling situations with hundreds or thousands of unknown model parameters.
