A Framework for Bayesian and Likelihood Approximations in Statistics

Abstract

Regular Bayesian and frequentist approximations in statistics are studied within a unified framework. In particular it is shown how some higher-order likelihood-based approximations arise from their Bayesian counterparts via an unsmoothing argument. This approach serves to shed new light on these formulae and to clarify relationships between Bayesian and frequentist inferences. For example, Bayesian interpretations of higher-order approximations can give insights into the acceptability, or otherwise, of these approximations from the point of view of ‘relevance’ to the actual data observed. Furthermore, this approach is a very natural one for the study of more general ‘nonregular’ problems, models for dependent data, and approximate conditional inference. For ease of exposition the development is in terms of a single real parameter. The main analytic development proceeds in terms of signed roots of log-density ratios.