Rejoinder.

I appreciate very much the comments of the four discussants, as well as those of numerous other individuals who made verbal comments about the paper. The discussants represent a variety of viewpoints in their attitude to Bayesian and frequentist statistics, and in the extent to which they accept decision theory as an appropriate framework for the assessment of statistical procedures. I believe it will help to set both the paper and the discussion in context if I present some personal opinions about how these diierent viewpoints interact. This will be followed by some more direct responses to the various points made by the discussants. Why do we need a new assessment of the theoretical properties of Bayesian proce-dures? It is clear that Bayesian methods have been outstandingly successful as practical application of statistics | the Valencia conferences themselves present ample evidence of that. Nevertheless, it seems to me that much of the recent expansion of the subject has been driven by computational advances, especially those using Markov chain Monte Carlo algorithms, rather than by fundamental advances in Bayesian thinking. This is especially true within the eld of hierarchical models. However, the old arguments have not gone away. Indeed, it seems to me that with the application of Bayesian methods to ever more complex problems, it becomes imperative to look more closely at the extent to which seemingly arbitrary choices in the speciication of a prior distribution aaect the results of the analysis. I have absolutely no disagreement with those Bayesians, presumably including Professor D'Agostini, who would always use genuine prior knowledge to formulate their prior distributions. It is simply my own observation that the proportion of Bayesian papers that actually do this is rather small. Hence the need for alternative, more \objective" (whatever that means) criteria for comparing Bayesian procedures. There are, nevertheless, diierent approaches that one could take to this task, so it becomes necessary to defend one based primarily on frequentist reasoning, and especially on decision theory. I do not believe there is anything controversial about the notion that a sequence of probability forecasts, produced through some statistical estimation procedure, should provide a good long-run representation of the actual outcome of those forecasts. Indeed, this idea lies at the heart of Phil Dawid's theory of prequential assessment, as represented for example by Dawid (1984) as well as Dawid and Seillier-Moiseiwitsch (1993). The real diiculty lies in the speciication …