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This chapter describes the central claims of the book. From a philosophical point of view, the book argues for an objective Bayesian interpretation of probability and an epistemic interpretation of causality, and claims that these offer a firm foundation for causal modelling. From the computational point of view, the book investigates the relationship between Bayesian nets and maximum entropy methods, and develops a general computational framework for probabilistic and causal reasoning.

Objective Bayesianism yields a justification of the causal Markov condition: in certain circumstances, the objective Bayesian net is just the causal net and so the causal net is an appropriate representation of rational degrees of belief. However, the resulting Bayesian net may not yield accurate enough predictions. This motivates a two-stage methodology for using Bayesian nets: first construct a causal net, then refine this net to better represent physical probability.

Shows how Bayesian thinking should make democratic outcomes so rationally compelling. Bayes's formula provides a mathematical expression for specifying exactly how we ought rationally to update our a priori beliefs in light of subsequent evidence, and the proposal is that voters are modelled in like fashion: votes, let us suppose, constitute (among other things) ‘reports’ of the voter's experiences and perceptions; further suppose that voters accord ‘evidentiary value’ to the reports they receive from one another through those votes; and further suppose that voters are rational, and that part and parcel of their being rational is being prepared to revise their opinions in light of further evidence (including evidence emanating from one another's votes‐cum‐reports). In this process, each of us treats our own experiences and perceptions as one source of evidence, and regards our own report as right; in that sense, we are perfectly sincere when we vote in a particular way, although we also acknowledge that our own experiences and perspectives are particular and peculiar, and hence our own perceptions are themselves inconclusive; because of that, voters striving to behave rationally should sincerely want to adjust their a priori beliefs in the light of all other experiences and perceptions that are reported at an election. Bayesian updating of that sort may well lead people who started out believing (and voting) one way to end up believing (and genuinely wanting implemented) the opposite way, just so long as sufficiently many votes‐cum‐reports point in that different direction; in other words, Bayesian reasoning can, and in politically typical cases ought to, provide people with a compelling reason to accede to the majority verdict. In this way, Bayesianism ‘rationalizes’ majority rule in a pretty strong sense; indeed if anything, it underwrites majoritarianism too strongly.

This is the last of four chapters on belief democracy, and discusses how to rationalize persistent opposition in the light of Wollheim's paradox (how can you vote one way, and then immediately accept the majority verdict in favour of the opposite view, when voting the first way is seen as asserting that it is true and acceding to the majority verdict is seen as accepting that the opposite is true), and Bayesian considerations that make it utterly irrational for anyone ever to continue disagreeing once everyone has voted. The first section demonstrates that none of the easy and obvious ways of extending (modifying) the Bayesian framework rationalize majoritarianism without derationalizing ongoing opposition. Next, various ways of moving beyond Bayes to overcome the paradox of persisting opposition are put forward; these include repudiating Bayesian reasoning altogether, assuming that disagreement betokens unreliability, assuming that smaller groups are inherently more reliable, and assuming that each election is different; none of these produce the required result. Finally, political arguments are put forward more explicitly to see whether they can rationalize ongoing opposition; these include the crucial role of opposition in democratic politics, proceduralism, denying propositional content, biased perception, strategic voting, segmented information pools, and different interests. In the light of what has been presented overall, the author puts forward his own explanation, which suggests that what is wrong with Bayesian models that ask us to update our beliefs in the light of others’ votes is that those votes mix facts and values, and only one of these (others’ assessments of the facts) can reasonably be taken into account in updating our beliefs; thus, it is the epistemic power of majorities when dealing with shared facts that underwrites the rationality of majority rule, but their lack of epistemic authority when it comes to matters of evaluations that underwrites the rationality of persisting opposition.

Probability functions over domains of variables are introduced, as are the various interpretations of probability, including the frequency, propensity, chance, and Bayesian interpretations. The Bayesian notion of chance as ultimate belief is also discussed.

This chapter discusses the scope and limits of Bayesianism. The first topics discussed are the math and philosophy embedded in Bayesianism and the standard objection to Bayesianism. The concept of likelihood in Bayesianism is examined, and a problem of the best-case strategy of dealing with nuisance parameters is studied. Finally, the main weakness of strong Bayesianism is identified.

This chapter has two purposes. First, it explores the connection between analogical reasoning and symmetry to provide a second justification for the argument form, independent of the one offered in chapter 7. The chapter proposes that good analogical arguments are sanctioned by norms of symmetry and Reflection, as expounded by van Fraassen. This idea is elaborated first in a nonprobabilistic setting and then in a probabilistic framework. It is argued that a psychological construal of the relationship between symmetry and probability, as championed by de Finetti and others, is inadequate. The second objective of the chapter is to integrate analogical arguments into a Bayesian model of theoretical confirmation by refining Salmon's idea that analogical arguments contribute to establishing non‐negligible prior probability for hypotheses.

This chapter argues that if the universe is governed by laws of nature, then particular laws of nature cannot be discovered by empirical science unless the meta‐theoretic conception of laws is correct. This gives us a powerful reason to embrace the meta‐theoretic conception. The key argument appeals to underdetermination of theory by evidence, but it avoids the pitfalls of familiar underdetermination arguments against realism. Several objections to the argument are considered, including one based on Bayesianism, one based on inference to the best explanation (IBE), one based on contextualist epistemology, and one that alleges that the only way to avoid inductive skepticism is to adopt an epistemology of science that allows for the discovery of laws of nature in spite of radical underdetermination of the laws by the available evidence. All these objections are found wanting.

This chapter begins with a summary of the classical sociological solution to the problem of order advanced by Talcott Parsons. The central flaw in the solution is simply that it lacks deliberative microfoundations; it is not clear how social norms interact with beliefs and desires, the two intentional states posited by standard decision theory. A solution is proposed, which involved positing “principles” as a set of intentional states associated direction with actions. A formal representation is developed, and the attractions of the model, from the standpoint of representing a variety of different social interaction types, are then touted.

This chapter introduces Bayes' Theorem, which is primarily concerned with probability. It discusses some of the different kinds of probability, such as statistical probability, before moving on to a discussion of probability axioms. Bayesianism, prior probability and simplicity, and countable additivity are also studied in the chapter.

This chapter discusses Bayesianism in statistics. The first section of the chapter is devoted to the First Bayesian Theory, which is immediately followed by a discussion of significance tests and the Second Bayesian Theory. Lindley's Paradox and the Neyman-Pearson Theory are examined in detail, along with the concept of priors and likelihood. The final portion of the chapter focuses on the Second Bayesian theory as logic.

This essay discusses the programme called ‘therapeutic Bayesianism’ from three abstract points of view: substantial, foundational, and meta-philosophical. It illustrates treatments of the ‘raven’ paradox and the puzzle of diverse data, and defends the propriety of certain idealizations. It criticises a meta-philosophical perspective that does not properly distinguish science from the philosophy of science, and overvalues the use of symbolic apparatus.

In “Scientific Knowledge,” Philip Kitcher challenges arguments that deny the truth of the theoretical claims of science, and he attempts to discover reasons for endorsing the truth of such claims. He suggests that the discovery of such reasons might succeed if we ask why anyone thinks that the theoretical claims we accept are true and then look for answers that reconstruct actual belief‐generating processes. To this end, Kitcher presents the “homely argument” for scientific truth, which claims that when a field of science is continually applied to yield precise predictions, then it is at least approximately true. He defends this approach and offers a supplementary account that gives more attention to detail. This account includes a historical aspect (a dependence on the previous conclusions of scientists) that must answer to skeptical challenges and a social aspect (the coordination of individuals in pursuit of specific knowledge‐related goals).

In ”Decision Theory and Epistemology,” Mark Kaplan finds it characteristic of orthodox Bayesians to hold that (I) for each person and each hypothesis she comprehends, there is a precise degree of confidence that person has in the truth of that proposition, and (II) no person can be counted as rational unless the degree of confidence assignment she thus harbors satisfies the axioms of the probability calculus. Kaplan's purpose is twofold. First, he aims to show that, as powerful as many criticisms are against orthodox Bayesianism, there is a credible kind of Bayesianism. Without appeal to idealization or false precision, it offers a substantive account of how the probability calculus constrains the (imprecise) opinions of actual persons and of how this account impinges on traditional epistemological concerns. Second, he aims to show how this Bayesianism finds a foundation in considerations concerning rational preference.

Bayesianism claims to provide a unified theory of epistemic and practical rationality based on the principle of mathematical expectation. In its epistemic guise it requires believers to obey the laws of probability. In its practical guise it asks agents to maximize their subjective expected utility. Joyce’s primary concern is Bayesian epistemology, and its five pillars: (1) people have beliefs and conditional beliefs that come in varying gradations of strength; (2) a person believes a proposition strongly to the extent that she presupposes its truth in her practical and theoretical reasoning; (3) rational graded beliefs must conform to the laws of probability; (4) evidential relationships should be analyzed subjectively in terms of relations among a person’s graded beliefs and conditional beliefs; (5) empirical learning is best modeled as probabilistic conditioning. Joyce explains each of these claims and evaluates some of the justifications that have been offered for them, including “Dutch book,” “decision-theoretic,” and “non-pragmatic” arguments for (3) and (5). He also addresses some common objections to Bayesianism, in particular the “problem of old evidence” and the complaint that the view degenerates into an untenable subjectivism. The essay closes by painting a picture of Bayesianism as an “internalist” theory of reasons for action and belief that can be fruitfully augmented with “externalist” principles of practical and epistemic rationality.