The interface between Bayesian and frequentist statistics

Statistical theory is often categorized as either "Bayesian" or "frequentist", and statisticians often self-identify in the same categories. During the development of theoretical statistics as a separate field in the twentieth century this categorisation led to a great deal of discussion, some of which was surprisingly bitter and antagonistic. With the development of several key results in the asymptotic theory of inference based on the likelihood function, it is becoming clear that the mathematical differences between Bayesian and frequentist methods are rather less important than the philosophical ones. Some of this work is based on efforts to construct priors which minimize the difference between the two approaches and some is based on an ongoing effort to develop so-called 'reference', or 'objective' or ;default' priors. Perhaps not surprisingly, even the correct terminology to be used in this setting has been the subject of debate!

I will give an overview of some of the asymptotic theory behind the development of approaches to constructing priors that minimize the differences between Bayesian and frequentist inference, with special attention to 'strong matching' priors that have been developed recently in joint work with Don Fraser and colleagues. The construction of these priors provides some insight into the exact points of departure between Bayesian and frequentist methods, at least from the mathematical point of view. The philosophical debate may well continue for some time.