How should hypotheses be evaluated, what is the role of evidence in that process, what are the most informative experiments to perform? 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 . . .