Interest in Bayesian methods has been growing rapidly in the statistical community. This seminar discusses the Bayesian philosophical approach and its relevance to practicing statisticians. The viewpoint I present is mine, although I draw from the work of many who have thought deeply about these issues.

I begin the seminar with a discussion of different viewpoints on probability. In my view, we have been barking up the wrong tree in our debate about what probability "really is." What probability "really is" is a mathematical structure. No one quarrels about the properties of the mathematical structure or the correctness of the mathematics used by any of the parties to the debate. The debate concerns what real-world phenomena it is "legitimate" to model with this mathematical framework. Phenomena modeled by probability can be arranged in a hierarchy of generality, from equipossible outcomes in problems with natural symmetries to frequencies of "random" events, to rational degrees of belief about arbitrary uncertain phenomena. Given this perspective, it seems to me unsupportable to take a categorical philosophical stance against ever applying probabilty theory to the third class of phenomena.

Similarly, I regard the dogmatic Bayesian who insists on "coherence" at all costs to be a bad decision theorist, unless (s)he REALLY cares about coherence more than getting a sensible answer to the problem at hand. Ultimately, then, we have to judge applications of probability theory by how well they help us solve the problem they were designed to solve. Adding the Reverend's legacy to our toolkit has pragmatic benefits whether or not we experience a religious conversion.

The remainder of the talk concerns these pragmatic benefits. The first benefit concerns relevance of the results obtained to the problem of interest. The frequentist draws inferences about data given unknown parameter, while the Bayesian updates beliefs on uncertain parameters given new data. In virtually all applications, the latter type of conclusion is more relevant. I go on to give a highly abbreviated overview of some of the most exciting new developments in Bayesian statistics. Particular attention is given to the use of hierarchical models to provide robust and efficient methods for application to high-dimensional models. I conclude with some remarks about robustness and a Bayesian perspective on the problem of diagnosing model adequacy.