The choice of sample size-a reply to the discussion

Abstract

There has been some confusion over the term 'Bayesian'. Without wishing to appear too much like Humpty-Dumpty, I use the term to describe an approach to inference and decision analysis that expresses uncertainty about an entity solely in terms of probability concerning that entity, assesses the outcomes of a decision through utility and then selects that decision that maximizes expected utility, the expectation using the inference probability. The discussants admit a probability density p(O) for the binomial parameter 0, and to that extent are Bayesian in their inference, but do not include a utility in their decision-making, and are thereby not overtly Bayesian. I hope that my definition would meet with the approval of all participants in the 5th International Conference on Bayesian Statistics (Bernardo et al., 1996), for example. One reason for adopting the Bayesian position is the desire to achieve coherence. The Shorter Oxford Dictionary defines 'coherence' as 'harmonious connexion of the several parts of a discourse, system etc., so that the whole hangs together'. We are discussing inference and decision-making. We require the procedure adopted with a sampe of size 10 to 'hang together' with that for size 20. It is known (Berger and Delampady, 1987) that the use of a P-value for testing a sharp, null hypothesis does not satisfy this requirement; 5% significance for n = 10 has not the same meaning as 5% with n = 20. Point estimation of a normal mean by the sample mean does, to a very good approximation, satisfy the requirement in one and two dimensions, but not in three or more. What precisely is meant by hangs together in the statistical context: what is the glue that makes one procedure fit, or cohere, with another? The usual glue for a discourse is logic, but logic only deals with truth or falsity and statisticians are concerned with uncertainty, so it is necessary to go beyond logic. The solution adopted is to postulate some basic requirements for the glue. For example, in the decision aspect of coherent behaviour, it is supposed that between any two decisions you have a preference, and that these preferences are transitive in that if d, is preferred to d2, and d2 to d3, then d1 is preferred to d3. Here the first two preferences are required to cohere with the third. Several basic requirements of this type are adopted as axioms and a mathematical analysis developed that allows us to produce coherent procedures in inference and decision analysis. The simplest set of axioms leads to the conclusion stated in my first paragraph: uncertainty must be described by probability, the worth of decisions by utility and the preferred decision selected by maximization of expected utility (MEU). The axioms can be weakened and the conclusions modified. For example, Walley (1991) established only lower and upper probabilities, rather than a single probability. Such complications, important as they are, are not the issue here. A more important point is that no axioms known to me lead, for example, to fuzzy logic, or to the use of tail area probabilities in hypothesis testing. Although our Bayesian conclusion is constructive in its adoption of MEU, it is liberal in its unrestricted use of probability and utility. Consequently it is often