Preface to the Third Edition
- How should hypotheses be evaluated, what is the role of evidence in that process, what are the most informative experiments to per-form? Questions such as these are ancient ones. They have been answered in various ways, often exciting lively controversy, not surprisingly in view of the important practical implications that different answers carry. Our approach to these questions, which we set out in this book, is the Bayesian one, based on the idea that valid inductive reasoning is reasoning according to the formal principles of probability.
- The Bayesian theory derives from the Memoir of the mathematician and divine, Thomas Bayes, which was published posthumously by his friend Richard Price in 1763. The principles set out by Bayes had considerable influence in scientific and philosophical circles, though worries about the status of the prior probabilities of scientific theories meant that the whole approach continued to be dogged by debate. And by the 1920s, an alternative approach, often called ‘Classical’, achieved dominance, due to powerful advocacy by R. A. Fisher and many other distinguished statisticians, and by Karl Popper and similarly distinguished philosophers. Most of the twentieth century was dominated by the classical approach, and in that period Bayesianism was scarcely taught in universities, except to disparage it, and Bayesians were widely dismissed as thoroughly misguided.
- But in recent years, there has been a sea-change, a paradigm shift. A search of the Web of Science database shows, during the 1980s. a regular trickle of around 200 articles published annually with the word or prefix ’Bayes’ in their titles. Suddenly, in 1991, this number shot up to 600 and by 1994 exceeded 800; by 2000 it had reached almost 1,400. (Williamson and Corfield. 2001, p. 3). This book was one of the first to present a comprehensive, philosophical case for the Bayesian approach to scientific reasoning and to show its superiority over the classical. Its first and second editions were published in 1989 and 1993, and from the figures quoted it is clear that the book anticipated a powerful and sophisticated resurgence of the once-dominant Bayesian approach.
- This new edition amends, updates, re-organizes, and seeks to make the subject more accessible. The text is intended to be self- contained. calling, in the main, on only elementary mathematical and statistical ideas. Nevertheless, some parts are more complex, and some more essential to the overall argument than others. Accordingly, we would suggest that readers who are not already familiar with mathematical probability but who wish to gain an initial understanding of the Bayesian approach, and to appreciate its power, adopt the following plan of attack.
- First, read Chapter 1, which sets the scene, as it were, with a brief historical overview of various approaches to scientific inference.
- Then, look at section a of Chapter 2, which gives the simple principles or axioms of the probability calculus, and section b, where there are some of the probability theorems that will be found useful in the scientific context: the central theorem here is Bayes’s theorem in its various forms.
- We then suggest that the reader look at the first few sections of Chapter 4, where Bayes’s theorem is applied to some simple reasoning patterns that arc found particularly when deterministic theorems are handled; this chapter also compares the Bayesian approach with some others, such as Popper’s well-known falsificationist methodology.
- Chapters 5 to 7 deal with non-deterministic, that is, statistical hypotheses, giving a critical exposition of the classical, or frequentist, methods that constitute the leading alternative to the Bayesian approach: the main classical ideas can be gleaned from sections a to d and f and g of Chapter 5.
- The final part of the mini-course we are suggesting is to examine Chapter 9, where some of the more widespread criticisms that have been levelled against the Bayesian approach are discussed (and rejected).
- There are some marked differences between this third edition and the preceding ones. For example, some of the objections to the Bayesian theory we considered in the second edition have not stood the test of time. There have also been changes of mind: one of the most prominent examples is the fact that now we accept the strength of de Finetti’s well-known arguments against countable additivity, and have accordingly dropped it as a generally valid principle. Other changes have been largely dictated by the desire to make this edition more compact and thereby more accessible. We hope that this indeed turns out to be the case.
Text Colour Conventions (see disclaimer)
- Blue: Text by me; © Theo Todman, 2020
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