Bayesian Statistics

An Introduction

PETER M. LEE

Preface to the First Edition

When I first learned a little statistics, I felt confused, and others I spoke to confessed that they had similar feelings. Not because the mathematics was difficult-most of that was a lot easier than pure mathematics-but because I found it difficult to follow the logic by which inferences were arrived at from data. It sounded as if the statement that a null hypothesis was rejected at the 5% level meant that there was only a 5% chance of that hypothesis was true, and yet the books warned me that this was not a permissible interpretation. Similarly, the statement that a 95% confidence interval for an unknown parameter ran from -2 to +2 sounded as if the parameter lay in that interval with 95% probability and yet I was warned that all I could say was that if I carried out similar procedures time after time then the unknown parameters would lie in the confidence intervals I constructed 95% of the time. It appeared that the books I looked at were not answering the questions that would naturally occur to a beginner, and that instead they answered some rather recondite questions which no-one was likely to want to ask.

Subsequently, I discovered that the whole theory had been worked out in very considerable detail in such books as Lehmann (1959 and 1986). But attempts such as those that Lehmann describes to put everything on a firm foundation raised even more questions. I gathered that the usual t test could be justified as a procedure that was “uniformly most powerful unbiased”, but I could only marvel at the ingenuity that led to the invention of such criteria for the justification of the procedure, while remaining unconvinced that they had anything sensible to say about a general theory of statistical inference. Of course, Lehmann and others with an equal degree of common sense were capable of developing more and more complicated constructions and exceptions so as to build up a theory that appeared to cover most problems without doing anything obviously silly, and yet the whole enterprise seemed reminiscent of the construction of epicycycle upon epicycle in order to preserve a theory of planetary motion based on circular motion; there seemed to be an awful lot of “adhockery”.

I was told that there was another theory of statistical inference, based ultimately on the work of the Rev. Thomas Bayes, a Presbyterian minister who ived from 1702 to 1761 whose key paper was published posthumously by his friend Richard Price as Bayes (1763) [more information about Bayes himself and his work can be found in Holland (1962), Todhunter (1865 and 1949) and Stigler (1986a); further information is now available in Bellhouse et al. (1988–1992), Dale (1991) and Edwards (1993).] However, I was warned that there was something not quite proper about this theory, because it depended on your personal beliefs and so was not objective. More precisely, it depended on taking some expression of your beliefs about an unknown quantity before the data was available (your “prior probabilities”) and modifying them in the light of the data (via the so-called “likelihood function”) to arrive at your “posterior probabilities” using the formulation that “posterior is proportional to prior times likelihood”. The standard, or “classical”, theory of statistical inference, on the other hand, was said to be objective, because it does not refer to anything corresponding to the Bayesian notion of “prior beliefs”. Of course, the fact that in this theory you sometimes looked for a 5% significance test and sometimes for a 0.1% signicance test, depending on what you thought about the different situations involved, was said to be quite a different matter.

I went on to discover that this theory could lead to the sorts of conclusions that I had naïvely expected to get from statistics when I first learned about it. Indeed, some lecture notes of Lindley’s [and subsequently his book, Lindley (1965)] and the pioneering book by Jeffreys (1939, 1948 and 1961) showed that if the statistician had “personal probabilities” that were of a certain conventional type then conclusions very like those in the elementary books I had first looked at could be arrived at, with the difference that a 95% confidence interval really did mean an interval in which the statistician was justified in thinking that there was a 95% probability of finding the unknown parameter. On the other hand, there was the further freedom to adopt other initial choices of personal beliefs and thus to arrive at different conclusions.

Over a number of years I taught the standard, classical, theory of statistics to a large number of students, most of whom appeared to have similar difficulties to those I had myself encountered in understanding the nature of the conclusions that this theory comes to. However, the mere fact that students have difficulty with a theory does not prove it wrong. More importantly, I found the theory did not improve with better acquaintance, and I went on studying Bayesian theory. It turned out that there were real differences in the conclusions arrived at by classical and Bayesian statisticians, and so the former was not just a special case of the latter corresponding to a conventional choice of prior beliefs. On the contrary, there was a strong disagreement between statisticians as to the conclusions to be arrived at in certain standard situations, of which I will cite three examples for now. One concerns a test of a sharp null hypothesis (for example a test that the mean of a distribution is exactly equal to zero), especially when the sample size was large. A second concerns the Behrens-Fisher problem, that is, the inferences that can be made about the difference between the means of two populations when no assumption is made about their variances. Another is the likelihood principle, which asserts that you can only take account of the probability of events that have actually occurred under various hypotheses, and not of events that might have happened but did not; this principle follows from Bayesian statistics and is contradicted by the classical theory. A particular case concerns the relevance of stopping rules, that is to say whether or not you are entitled to take into account the fact that the experimenter decided when to stop experimenting depending on the results so far available rather than having decided to use a fixed sample size all along. The more I thought about all these controversies, the more I was convinced that the Bayesians were right on these disputed issues.

At long last, I decided to teach a third-year course on Bayesian statistics in the University of York, which I have now done for a few years. Most of the students who took the course did find the theory more coherent than the classical theory they had learned in the first course on mathematical statistics they had taken in their second year, and I became yet more clear in my own mind that this was the right way to view statistics. I do, however, admit that there are topics (such as nonparametric statistics) which are difficult to fit into a Bayesian framework.

A particular difficulty in teaching this course was the absence of a suitable book for students who were reasonably well prepared mathematically and already knew some statistics, even if they knew nothing of Bayes apart from Bayes’ theorem. I wanted to teach them more, and to give more information about the incorporation of real as opposed to conventional prior information, than they could get from Lindley (1965), but I did not think they were well enough prepared to face books like Box and Tiao (1973) or Berger (1985), and so I found that in teaching the course I had to get together material from a large number of sources, and in the end found myself writing this book. It seems less and less likely that students in mathematics departments will be completely unfamiliar with the ideas of statistics, and yet they are not (so far) likely to have encountered Bayesian methods in their first course on statistics, and this book is designed with these facts in mind. It is assumed that the reader has a knowledge of calculus of one and two variables and a fair degree of mathematical maturity, but most of the book does not assume a knowledge of linear algebra. The development of the text is self-contained, but from time to time the contrast between Bayesian and classical conclusions is pointed out, and it is supposed that in most cases the reader will have some idea as to the conclusion that a classical statistician would come to, although no very detailed knowledge of classical statistics is expected. It should be possible to use the book as a course text for final year undergraduate or beginning graduate students or for self study for those who want a concise account of the way in which the Bayesian approach to statistics develops and the contrast between it and the conventional approach. The theory is built up step by step, rather than doing everything in the greatest generality to start with, and important notions such as sufficiency are brought out of a discussion of the salient features of specific examples.

I am indebted to Professor R.A. Cooper for helpful comments on an earlier draft of this book, although of course he cannot be held responsible for any errors in the final version.

Peter M. Lee

30 March 1988
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