Online research in philosophy

In the past, few mainstream epistemologists have endorsed Bayesian epistemology, feeling that it fails to capture the complex structure of epistemic cognition. The defenders of Bayesian epistemology have tended to be probability theorists rather than epistemologists, and I have always suspected they were more attracted by its mathematical elegance than its epistemological realism. But recently Bayesian epistemology has gained a following among younger mainstream epistemologists. I think it is time to rehearse some of the simpler but still quite devastating objections to Bayesian epistemology. Most of these objections are familiar, but have never been adequately addressed by the Bayesians.

Objective Bayesianism has been criticised on the grounds that objective Bayesian updating, which on a finite outcome space appeals to the maximum entropy principle, differs from Bayesian conditionalisation. The main task of this paper is to show that this objection backfires: the difference between the two forms of updating reflects negatively on Bayesian conditionalisation rather than on objective Bayesian updating. The paper also reviews some existing criticisms and justifications of conditionalisation, arguing in particular that the diachronic Dutch book justification fails because diachronic Dutch book arguments are subject to a reductio: in certain circumstances one can Dutch book an agent however she changes her degrees of belief . One may also criticise objective Bayesianism on the grounds that its norms are not compulsory but voluntary, the result of a stance. It is argued that this second objection also misses the mark, since objective Bayesian norms are tied up in the very notion of degrees of belief.

Inference to the Best Explanation (IBE) and Bayesianism are our two most prominent theories of scientific inference. Are they compatible? Van Fraassen famously argued that they are not, concluding that IBE must be wrong since Bayesianism is right. Writers since then, from both the Bayesian and explanationist camps, have usually considered van Fraassen's argument to be misguided, and have plumped for the view that Bayesianism and IBE are actually compatible. I argue that van Fraassen's argument is actually not so misguided, and that it causes more trouble for compatibilists than is typically thought. Bayesianism in its dominant, subjectivist form, can only be made compatible with IBE if IBE is made subservient to conditionalization in a way that robs IBE of much of its substance and interest. If Bayesianism and IBE are to be fit together, I argue, a strongly objective Bayesianism is the preferred option. I go on to sketch this objectivist, IBE-based Bayesianism, and offer some preliminary suggestions for its development.

Bayesianism is a collection of positions in several related fields, centered on the interpretation of probability as something like degree of belief, as contrasted with relative frequency, or objective chance. However, Bayesianism is far from a unified movement. Bayesians are divided about the nature of the probability functions they discuss; about the normative force of this probability function for ordinary and scientific reasoning and decision making; and about what relation (if any) holds between Bayesian and non-Bayesian concepts.

The proportional weight view in epistemology of disagreement generalizes the equal weight view and proposes that we assign to judgments of different people weights that are proportional to their epistemic qualifications. It is shown that if the resulting degrees of confidence are to constitute a probability function, they must be the weighted arithmetic means of individual degrees of confidence, while if the resulting degrees of confidence are to obey the Bayesian rule of conditionalization, they must be the weighted geometric means of individual degrees of confidence. The double bind entails that the proportional weight view (and its moderate adjustment in favor of one’s own judgment) is inconsistent with Bayesianism.

This is a review essay about David Corfield and Jon Williamson's anthology Foundations of Bayesianism. Taken together, the fifteen essays assembled in the book assess the state of the art in Bayesianism. Such an assessment is timely, because decision theory and formal epistemology have become disciplines that are no longer taught on a routine basis in good philosophy departments. Thus we need to ask: Quo vadis, Bayesianism? The subjects of the articles include Bayesian group decision theory, approaches to the concept of probability, Bayesian approaches in the philosophy of mathematics, reflections on the relationship between causation and probability, the Independence axiom, and a range of criticisms of Bayesianism, among other subjects. While critical of some of the arguments presented in the articles, this review recommends Corfield and Williamson's volume to anyone who is trying to stay abreast of Bayesian research.

Many philosophers argue that Bayesian epistemology cannot help us with the traditional Humean problem of induction. I argue that this view is partially but not wholly correct. It is true that Bayesianism does not solve Hume’s problem, in the way that the classical and logical theories of probability aimed to do. However I argue that in one important respect, Hume’s sceptical challenge cannot simply be transposed to a probabilistic context, where beliefs come in degrees, rather than being a yes/no matter.

Foundations of Bayesianism is an authoritative collection of papers addressing the key challenges that face the Bayesian interpretation of probability today. Some of these papers seek to clarify the relationships between Bayesian, causal and logical reasoning. Others consider the application of Bayesianism to artificial intelligence, decision theory, statistics and the philosophy of science and mathematics. The volume includes important criticisms of Bayesian reasoning and also gives an insight into some of the points of disagreement amongst advocates of the Bayesian approach. The upshot is a plethora of new problems and directions for Bayesians to pursue.The book will be of interest to graduate students or researchers who wish to learn more about Bayesianism than can be provided by introductory textbooks to the subject. Those involved with the applications of Bayesian reasoning will find essential discussion on the validity of Bayesianism and its limits, while philosophers and others interested in pure reasoning will find new ideas on normativity and the logic of belief.

Bayesian epistemology addresses epistemological problems with the help of the mathematical theory of probability. It turns out that the probability calculus is especially suited to represent degrees of belief (credences) and to deal with questions of belief change, confirmation, evidence, justification, and coherence. Compared to the informal discussions in traditional epistemology, Bayesian epis- temology allows for a more precise and fine-grained analysis which takes the gradual aspects of these central epistemological notions into account. Bayesian epistemology therefore complements traditional epistemology; it does not re- place it or aim at replacing it.

Bayesianism is our leading theory of uncertainty. Epistemology is defined as the theory of knowledge. So “Bayesian Epistemology” may sound like an oxymoron. Bayesianism, after all, studies the properties and dynamics of degrees of belief, understood to be probabilities. Traditional epistemology, on the other hand, places the singularly non-probabilistic notion of knowledge at centre stage, and to the extent that it traffics in belief, that notion does not come in degrees. So how can there be a Bayesian epistemology?