The 1988 Wald Memorial Lectures: The Present Position in Bayesian Statistics

Abstract

The first five sections of the paper describe the Bayesian paradigm for statistics and its relationship with other attitudes towards inference. Section 1 outlines Wald's major contributions and explains how they omit the vital consideration of coherence. When this point is included the Bayesian view results, with the main difference that Waldean ideas require the concept of the sample space, whereas the Bayesian approach may dispense with it, using a probability distribution over parameter space instead. Section 2 relates statistical ideas to the problem of inference in science. Scientific inference is essentially the passage from observed, past data to unobserved, future data. The roles of models and theories in doing this are explored. The Bayesian view is that all this should be accomplished entirely within the calculus of probability and Section 3 justifies this choice by various axiom systems. The claim is made that this leads to a quite different paradigm from that of classical statistics and, in particular, problems in the latter paradigm cease to have importance within the other. Point estimation provides an illustration. Some counter-examples to the Bayesian view are discussed. It is important that statistical conclusions should be usable in making decisions. Section 4 explains how the Bayesian view achieves this practicality by introducing utilities and the principle of maximizing expected utility. Practitioners are often unhappy with the ideas of basing inferences on one number, probability, or action on another, an expectation, so these points are considered and the methods justified. Section 5 discusses why the Bayesian viewpoint has not achieved the success that its logic suggests. Points discussed include the relationship between the inferences and the practical situation, for example with multiple comparisons; and the lack of the need to confine attention to normality or the exponential family. Its extensive use by nonstatisticians is documented. The most important objection to the Bayesian view is that which rightly says that probabilities are hard to assess. Consequently Section 6 considers how this might be done and an attempt is made to appreciate how accurate formulae like the extension of the conversation, the product law and Bayes rule are in evaluating probabilities.