In Defense of Objective Bayesianism covers a vast amount of ground in its articulation and defense of Jon Williamson’s atypical version of objective Bayesianism (hereafter OB). Following the introduction, chapters 2 and 3 articulate and motivate Williamson’s OB, while chapter 4 produces an ambitious line of argument against subjective Bayesianism, namely, denying the efficacy of the widely accepted Bayesian conditionalization method for belief updating. From chapters 5 through 9, Williamson examines the connections of his OB to the following themes: propositional and predicate languages; computational tools; the semantics of probabilistic logic; the problem of judgment aggregation; and richer languages. Of course, the review below falls short of accounting for all of Williamson’s interesting — and sometimes provocative — claims. Rather, this review highlights the ways in which Williamson’s OB is atypical, and notes the main claims in its defense found in chapters 2 through 4.
Bayesian epistemology is typically characterized by three core claims: (i) beliefs come in degrees; (ii) a rational agent A should conform the degree to which she believes that p to the degree to which her total evidence E makes p likely to be true; and (iii) A’s degree of belief that p should be updated through Bayesian conditionalization whenever relevant new evidence e is added to E.
Within this typical characterization, the floor is divided between objective Bayesians—the ones claiming that prior degrees of belief are fully determined by E — and subjective Bayesians — the ones claiming that prior degrees of belief are largely up to A. (‘Prior’ here means prior to the addition of e.) Though Williamson’s book defends a version of OB, it avoids this typical characterization. His book, as he puts it, “develops a version of objective Bayesianism that is not updated by conditionalization, and where probabilities are not fully determined by evidence — they are not always fully determined and, where they are, they are determined by more than evidence alone” (2). This atypical characterization is explained, defended, and expanded throughout the book.
Williamson’s OB is defined by three norms —Probability, Calibration, and Equivocation norms — each representing a constraint on A’s rational beliefs. He explains:
The Probability norm says that the strengths of an agent’s beliefs should be representable by a probability function defined over the sentences of the agent’s language. The Calibration norm says that this probability function should fit with the agent’s evidence. The Equivocation norm says that, to the extent that evidence leaves open a range of evidentially compatible probability functions, the agent’s degrees of belief should be representable by some such function that equivocates to a sufficient extent between the basic propositions expressible in her language. (p. 2)
I note three atypical aspects of Williamson’s OB. First, agent A’s total body of evidence E includes all that A grants, justifiably or not, and not all that that A knows or justifiably believes. This is a striking feature since what A grants can vary according to A’s operating context. Qua parent, for example, A might take his daughter’s testimony for granted; qua jury member, however, A might not. Second, since A’s language also varies according to operating contexts, and since A’s language at least partly determines A’s information about the world, the propositions available as putative beliefs for A may vary according to operating contexts as well. Third, since A’s E may leave open a range of evidentially compatible probability functions, it may turn out that, in circumstance C, there are “several equally rational degrees of belief from which the agent should choose one arbitrarily” (9). This may occur without pains of irrationality since, for Williamson, the rationality of A’s belief that p may vary according to A’s purposes in forming beliefs. Of these three aspects, at least the first and third seem controversial. (See chapters 1 and 2 for discussion.)
Chapter 3 discusses the motivation behind each norm. The overarching theme is that these norms “are forced upon us by the uses to which the strengths of our beliefs are put” (31). In other words, if A’s purpose in forming beliefs is that of forming true beliefs, then A’s beliefs are rational to the extent that they are governed by OB’s norms. This approach yields defenses of each norm based, respectively, on betting considerations, long-run betting considerations, and caution considerations. Here I note that these defenses hinge on an interpretation of the meaning of our concept of “rational degree of belief.” What this means, Williamson suggests, is that A should believe that p to the same degree as A would, roughly, be willing to bet on p being true.
Chapter 4 neatly turns a common criticism of OB on its head. It is common to argue that OB’s maximum entropy updating process is incompatible with Bayesian conditionalization; in many clear cases, it is put forth, each determines a different degree of belief. Williamson confirms that this is the case, but argues that this counts against Bayesian conditionalization instead. His main argument is a case study showing that Bayesian conditionalization fails to deliver the intuitively desired results more often than OB’s maximum entropy. This is as controversial as it is ingenious, and the matter is far from settled. Williamson’s case studies are nonetheless probing and worth close examination. I note that, further into the chapter, He also produces independent arguments against Bayesian conditionalization which, if sound, would show subjective Bayesianism to be intolerably permissive in its priors.
As noted above, chapters 5 through 9 examine the connections of Williamson’s OB to propositional and predicate languages, computational tools, the semantics of probabilistic logic, the problem of judgment aggregation, and richer languages. The book’s tenth and last chapter brings an interesting self-assessment and suggests ways in which further research could expand the arguments in the book.
In Defense of Objective Bayesianism is a very welcome book for anyone interested in Bayesianism — perhaps one of the hottest topics in contemporary epistemology. Epistemologists should find the first 4 chapters particularly probing, and mathematicians (I suspect, but can only suspect) should find all of it enriching. The book requires substantial familiarity with probability theory, however, and thus would be undesirable as an introduction.
Luis Oliveira is a graduate student in philosophy at the University of Massachusetts, Amherst. He is mainly interested in the traditional epistemological concepts of knowledge, evidence, and justification, and can be contacted at firstname.lastname@example.org.