From Bayesianism to the Epistemic View of Mathematics

‘Subjective probability: the real thing’ is the last book written by the late Richard Jeffrey, a key proponent of the Bayesian interpretation of probability. Bayesians hold that probability is a mental notion: saying that the probability of rain is 0.7 is just saying that you believe it will rain to degree 0.7. Degrees of belief are themselves cashed out in terms of bets—in this case you consider 7 : 3 to be fair odds for a bet on rain. There are two extreme Bayesian positions. Strict Subjectivists think that an agent can adopt whatever degrees of belief she likes, as long as they satisfy the axioms of probability. Thus your degree of belief in rain and degree of belief in no rain must sum to one but are otherwise unconstrained. At the other extreme, objectivists claim that an agent’s background knowledge considerably narrows down the choice of appropriate degrees of belief. In particular, if you know only that the frequency of rain is 0.7 then you should believe it will rain to degree 0.7; if you know absolutely nothing about the weather then you should set your degree of belief in rain to be 0.5; in neither of these cases is there room for subjective choice of degree of belief. In this book, Jeffrey advocates what is sometimes called empirically-based subjectivism, a position that lies between the two extremes of strict subjectivism and objectivism. According to this position, knowledge of frequencies constrains degree of belief, but lack of knowledge does not impose any constraints, so that if you know nothing about the weather you may adopt any degree of belief in rain you like. The aim of the book isn’t so much to justify this point of view as to provide a comprehensive exposition of probability theory from the perspective that it offers. The book succeeds admirably: Jeffrey presents a broad range of standard topics concerning Bayesianism, including the betting interpretation of degrees of belief, a discussion of objective chance, the application of Bayesianism to scientific reasoning, conditionalisation, expectation, exchangeability and decision theory. Naturally much of the discussion of these topics focuses on Jeffrey’s own multifarious contributions to the subject. For example, Jeffrey devotes much more space to his own version of conditionalisation than to the more standard Bayesian conditionalisation; he only discusses the version of decision theory