Jon Williamson and David Corfield Introduction: Bayesianism into the 21st Century

Bayesian theory now incorporates a vast body of mathematical, statistical and computational techniques that are widely applied in a panoply of disciplines, from artificial intelligence to zoology. Yet Bayesians rarely agree on the basics, even on the question of what Bayesianism actually is. This book is about the basics — about the opportunities, questions and problems that face Bayesianism today. So what is Bayesianism, roughly? Most Bayesians maintain that an individual’s degrees of belief ought to obey the axioms of the probability calculus. If, for example, you believe to degree 0.4 that you will be rained on tomorrow, then you should also believe that you will not be rained on tomorrow to degree 0.6. Most Bayesians also maintain that an individual’s degrees of belief should take prior knowledge and beliefs into account. According to the Bayesian conditionalisation principle, if you come to learn that you will be in Manchester tomorrow (m) then your degree of belief in being rained on tomorrow (r) should be your previous conditional belief on r given m: pt+1(r) = pt(rjm). By Bayes’ theorem this can be rewritten pt(mjr)pt(r)=pt(m).1 Although Bayesianism was founded in the eighteenth century by Thomas Bayes2 and developed in the nineteenth century by Laplace,3 it was not until well into the twentieth century that Frank Ramsey4 and Bruno de Finetti5 provided credible justifications for the degree of belief interpretation of probability, in the shape of their Dutch book arguments. A Dutch book argument aims to show that if an agent bets according to her degrees of belief and these degrees are not probabilities, then the agent can be made to lose money whatever the outcome of the events on which she is betting. Already by this stage we see disagreement as to the nature of Bayesianism, centring on the issue of objectivity. De Finetti was a strict subjectivist: he believed that probabilities only represent degrees of rational belief, and that an agent’s belief function is rational just when it is a probability function — no further constraints need to be satisfied.6 Ramsey, on the other hand, was a pluralist in that he also accepted objective frequencies. Further, he advocated a kind of calibration between degrees of belief and frequencies: