Probabilistic models have much to offer to philosophy. We continually receive information from a variety of sources: from our senses, from witnesses, from scientific instruments. When considering whether we should believe this information, we assess whether the sources are independent, how reliable they are, and how plausible and coherent the information is. Bovens and Hartmann provide a systematic Bayesian account of these features of reasoning. Simple Bayesian Networks allow us to model alternative assumptions about the nature of the (...) information sources. Measurement of the coherence of information is a controversial matter: arguably, the more coherent a set of information is, the more confident we may be that its content is true, other things being equal. The authors offer a new treatment of coherence which respects this claim and shows its relevance to scientific theory choice. Bovens and Hartmann apply this methodology to a wide range of much discussed issues regarding evidence, testimony, scientific theories, and voting. Bayesian Epistemology is an essential tool for anyone working on probabilistic methods in philosophy, and has broad implications for many other disciplines. (shrink)

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 (...) class='Hi'>epistemology therefore complements traditional epistemology; it does not re- place it or aim at replacing it. (shrink)

‘Bayesian epistemology’ became an epistemological movement in the 20th century, though its two main features can be traced back to the eponymous Reverend Thomas Bayes (c. 1701-61). Those two features are: (1) the introduction of a formal apparatus for inductive logic; (2) the introduction of a pragmatic self-defeat test (as illustrated by Dutch Book Arguments) for epistemic rationality as a way of extending the justification of the laws of deductive logic to include a justification for the laws of (...) inductive logic. The formal apparatus itself has two main elements: the use of the laws of probability as coherence constraints on rational degrees of belief (or degrees of confidence) and the introduction of a rule of probabilistic inference, a rule or principle of conditionalization. (shrink)

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? (shrink)

Bayesian epistemology provides a formal framework within which concepts in traditional epistemology, in particular concepts relating to the justification of our beliefs, can be given precise definitions in terms of probability. The Bayesian approach has contributed clarity and precision to a number of traditional issues. A salient example is the recent embedding of the so-called coherentist theory of epistemic justification in a Bayesian framework shedding light on the relation between coherence and truth as well as (...) on the concept of coherence itself. Starting with the early work of Condorcet, the calculus of probability has proved to be a useful tool in the study of social aspects of knowledge as it is pursued in social epistemology. (shrink)

'Fundamentals of Bayesian Epistemology' provides an accessible introduction to the key concepts and principles of the Bayesian formalism. This volume introduces degrees of belief as a concept in epistemology and the rules for updating degrees of belief derived from Bayesian principles.--.

Bayesian Epistemology is a general framework for thinking about agents who have beliefs that come in degrees. Theories in this framework give accounts of rational belief and rational belief change, which share two key features: (i) rational belief states are represented with probability functions, and (ii) rational belief change results from the acquisition of evidence. This dissertation focuses specifically on the second feature. I pose the Evidence Question: What is it to have evidence? Before addressing this question we (...) must have an understanding of Bayesian Epistemology. The first chapter argues that we should understand Bayesian Epistemology as giving us theories that are evaluative and not action-guiding. I reach this verdict after considering the popular ‘ought’-implies-‘can’ objection to Bayesian Epistemology. The second chapter argues that it is important for theories in Bayesian Epistemology to answer the Evidence Question, and distinguishes between internalist and externalist answers. The third and fourth chapters present and defend a specific answer to the Evidence Question. The account is inspired by reliabilist accounts of justification, and attempts to understand what it is to have evidence by appealing solely to considerations of reliability. Chapter 3 explains how to understand reliability, and how the account fits with Bayesian Epistemology, in particular, the requirement that an agent’s evidence receive probability 1. Chapter 4 responds to objections, which maintain that the account gives the wrong verdict in a variety of situations including skeptical scenarios, lottery cases, scientific cases, and cases involving inference. After slight modifications, I argue that my account has the resources to answer the objections. The fifth chapter considers the possibility of losing evidence. I show how my account can model these cases. To do so, however, we require a modification to Conditionalization, the orthodox principle governing belief change. I present such a modification. The sixth and seventh chapters propose a new understanding of Dutch Book Arguments, historically important arguments for Bayesian principles. The proposal shows that the Dutch Book Arguments for implausible principles are defective, while the ones for plausible principles are not. The final chapter is a conclusion. (shrink)

'Fundamentals of Bayesian Epistemology' provides an accessible introduction to the key concepts and principles of the Bayesian formalism. Volume 2 introduces applications of Bayesianism to confirmation and decision theory, then gives a critical survey of arguments for and challenges to Bayesian epistemology.--.

Inductive logic admits a variety of semantics (Haenni et al., 2011, Part 1). This paper develops semantics based on the norms of Bayesian epistemology (Williamson, 2010, Chapter 7). §1 introduces the semantics and then, in §2, the paper explores methods for drawing inferences in the resulting logic and compares the methods of this paper with the methods of Barnett and Paris (2008). §3 then evaluates this Bayesian inductive logic in the light of four traditional critiques of inductive (...) logic, arguing (i) that it is language independent in a key sense, (ii) that it admits connections with the Principle of Indifference but these connections do not lead to paradox, (iii) that it can capture the phenomenon of learning from experience, and (iv) that while the logic advocates scepticism with regard to some universal hypotheses, such scepticism is not problematic from the point of view of scientific theorising. (shrink)

In this article applied and theoretical epistemologies benefit each other in a study of the British legal case of R. vs. Clark. Clark's first infant died at 11 weeks of age, in December 1996. About a year later, Clark had a second child. After that child died at eight weeks of age, Clark was tried for murdering both infants. Statisticians and philosophers have disputed how to apply Bayesian analyses to this case, and thereby arrived at different judgments about it. (...) By dwelling on this applied case, I make theoretical gains: clarifying and defending pragmatic principles of inference that are important for estimating key probabilities in a range of cases. Then, partly by drawing on such principles, and uncovering overlooked data on post-partum psychosis, I make applied gains: improving the rationality of judgments about the Sally Clark case in particular, judgments important to future similar cases. (shrink)

Bayesian epistemology tells us with great precision how we should move from prior to posterior beliefs in light of new evidence or information, but says little about where our prior beliefs come from. It offers few resources to describe some prior beliefs as rational or well-justified, and others as irrational or unreasonable. A different strand of epistemology takes the central epistemological question to be not how to change one’s beliefs in light of new evidence, but what reasons (...) justify a given set of beliefs in the first place. We offer an account of rational belief formation that closes some of the gap between Bayesianism and its reason-based alternative, formalizing the idea that an agent can have reasons for his or her (prior) beliefs, in addition to evidence or information in the ordinary Bayesian sense. Our analysis of reasons for belief is part of a larger programme of research on the role of reasons in rational agency (Dietrich and List, Nous, 2012a, in press; Int J Game Theory, 2012b, in press). (shrink)

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This paper distinguishes between "descriptive" and "normative" conceptions of Bayesian principles of rationality, both in the context of inference and in the context of decision (which of course are not unrelated). I emphasize an idea according to which, "You have to work with what you have to work with" - that is, that rationality is a relation among old beliefs, new information, and new beliefs (in the case of inference) and among beliefs, desires, preferences, and choices (in the case (...) of decision). According to this conception of rationality, one's current beliefs and desires are not themselves subject to evaluation as to their rationality (except for some minimal, basically logical and "coherentist" constraints). From this perspective, rationality is about how we move from old beliefs (whatever they are) to new beliefs when confronted with evidence, and about how our preferences are structured given what we believe and what we want (whatever we currently happen to believe and want). I present some formal details of this perspective and discuss several criticisms of it. (shrink)

In this chapter we draw connections between two seemingly opposing approaches to probability and statistics: evidential probability on the one hand and objective Bayesian epistemology on the other.

Much contemporary epistemology is informed by a kind of confirmational holism, and a consequent rejection of the assumption that all confirmation rests on experiential certainties. Another prominent theme is that belief comes in degrees, and that rationality requires apportioning one's degrees of belief reasonably. Bayesian confirmation models based on Jeffrey Conditionalization attempt to bring together these two appealing strands. I argue, however, that these models cannot account for a certain aspect of confirmation that would be accounted for in (...) any adequate holistic confirmation theory. I then survey the prospects for constructing a formal epistemology that better accommodates holistic insights. (shrink)

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. (shrink)

Synthese 156 (3) (2007). Special issue ed. with Luc Bovens. With contributions by Max Albert, Branden Fitelson, Dennis Dieks, Igor Douven and Wouter Meijs, Alan Hájek, Colin Howson, James Joyce, and Patrick Suppes.

Hinge epistemology and Bayesianism are two prominent approaches in contemporary epistemology, but the relationship between these approaches has not been systematically studied. This paper formalizes the central commitments of hinge epistemology in a Bayesian framework and argues for the following two theses: (1) many of the types of claims that are treated as paradigmatic hinges in the hinge epistemology literature, such as the claim that there exists an external world of physical objects, are not capable (...) of enabling rational inquiry, even though this is typically regarded as a central property of hinges; (2) the standard Bayesian story of how rational inquiry proceeds is incorrect or at best incomplete. (shrink)

A common objection to theories of vagueness based on fuzzy logics centres on the idea that assigning a single numerical degree of truth -- a real number between 0 and 1 -- to each vague statement is excessively precise. A common objection to Bayesian epistemology centres on the idea that assigning a single numerical degree of belief -- a real number between 0 and 1 -- to each proposition is excessively precise. In this paper I explore possible parallels (...) between these objections. In particular I argue that the only good objection along these lines to fuzzy theories of vagueness does not translate into a good objection to Bayesian epistemology. An important part of my argument consists in drawing a distinction between two different notions of degree of belief, which I call dispositional degree of belief and epistemic degree of belief. (shrink)

In everyday matters, as well as in law, we allow that someone’s reasons can be causes of her actions, and often are. That correct reasoning accords with Bayesian principles is now so widely held in philosophy, psychology, computer science and elsewhere that the contrary is beginning to seem obtuse, or at best quaint. And that rational agents should learn about the world from energies striking sensory inputs nerves in people—seems beyond question. Even rats seem to recognize the difference between (...) correlation and causation,1 and accordingly make different inferences from passive observation than from interventions. A few statisticians aside,” so do most of us. To square these views with the demands of computability, increasing numbers of psychologists and others have embraced a particular formalization, causal Bayes nets, as an account of human reasoning about and to causal connections.111 Such structures can be used by rational agents, including humans in so far as they are rational, to have degrees of belief in various conceptual contents, which they use to reason to expectations, which are realized or defeated by sensory inputs, which cause them to change their degrees of belief in other contents in accord with Bayes Rule, or some generalization of it. How is all of this supposed to be carried out? l. Representing Causal Structures The causal Bayes net framework adopted by a growing number of psychologists goes like this: Our representations of causal relations are captured in a graphical causal. (shrink)

Bayesian epistemology has close connections to inductive reasoning, accepting the view that inductive inferences should be analyzed in terms of epistemic probabilities. An important precept of Bayesian epistemology is the dynamics of belief change, with change in belief resulting from updating procedures based on new evidence. The inductive relations between evidence E and hypotheses or theories H are essential, particularly the notions of plausibility, confirmation, and acceptability, which are critical but subject to several difficulties. As a (...) non-deductive process, Bayesian reasoning cannot itself be subjected to strict deductive logic, but it can take advantage of an enabling logical environment. The present paper proposes that paraconsistent and paracomplete logics can be helpful for some questions of Bayesian epistemology, even to the point of being relevant in developing a legitimate paraconsistent Bayesian epistemology. By developing a novel probability theory based on the Logic of Evidence and Truth ( $${LET}_{F}$$ ), a logic that deals with evidence for or against a judgment, including contradictory or missing evidence, we allow for the possibility of quantifying the degree of evidence attributed to a proposition through novel probability measures. We illustrate, through examples, some ways to address challenging problems in the area by using paraconsistent and paracomplete paradigms. This topic is of significant interest not only in the philosophy of science but also in Artificial Intelligence and other emerging trends such as probabilistic networks and related models (see Haenni et al. Probabilistic logics and probabilistic networks, volume 350, Springer, Berlin, 2010). (shrink)

Bayesian epistemology tells us with great precision how we should move from prior to posterior beliefs in light of new evidence or information, but says little about where our prior beliefs come from. It o¤ers few resources to describe some prior beliefs as rational or well-justified, and others as irrational or unreasonable. A di¤erent strand of epistemology takes the central epistemological question to be not how to change one's beliefs in light of new evidence, but what reasons (...) justify a given set of beliefs in the first place. We o¤er an account of rational belief formation that closes some of the gap between Bayesianism and its reason-based alternative, formalizing the idea that an agent can have reasons for his or her (prior) beliefs, in addition to evidence or information in the ordinary Bayesian sense. Our analysis of reasons for belief is part of a larger programme of research on the role of reasons in rational agency. (shrink)

The papers in this collection were presented at a workshop on Bayesian Epistemology at the 26th International Wittgenstein Symposium in Kirchberg, Austria (August 4–7, 2003), at a workshop on Philosophy and Probability at the conference GAP5 in Bielefeld, Germany (September 20–22, 2003), at a workshop on Bayesian Epistemology at the Centre for Philosophy of Natural and Social Science, London School of Economics and Political Science in London, UK (June 28, 2004), or at the seminar of the (...) research group Philosophy, Probability and Modeling, Center for Junior Research Fellows, University of Konstanz, Germany. We acknowledge the generous support for these events by the Austrian Ludwig Wittgenstein Society, the Gesellschaft für analytische Philosophie e.V., the Centre for Philosophy of Natural and Social Science, the Alexander von Humboldt Foundation, the Federal Ministry of Education and Research and the Program for the Investment in the Future (ZIP) of the German Government through a Sofja Kovalevskaja Award. We thank the organizers of the conferences in Kirchberg and Bielefeld for their support, the authors and referees of the papers for their work, and John Symons for his interest in dedicating a special issue of this journal to Bayesian Epistemology. (shrink)

Book Review of Luc Bovens and Stephan Hartmann *Bayesian Epistemology* by Erik J. Olsson.

English abstract: This paper discusses the delicate relationship between traditional epistemology and the increasingly influential probabilistic (or ‘Bayesian’) approach to epistemology. The paper introduces some of the key ideas of probabilistic epistemology, including credences or degrees of belief, Bayes’ theorem, conditionalization, and the Dutch Book argument. The tension between traditional and probabilistic epistemology is brought out by considering the lottery and preface paradoxes as they relate to rational (binary) belief and credence respectively. It is then (...) argued that this tension can be alleviated by rejecting the requirement that rational (binary) beliefs must be consistent and closed under logical entailment. Instead, it is suggested that this logical requirement applies to a different type of binary propositional attitude, viz. acceptance. (shrink)

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. (shrink)

Book Review of Luc Bovens and Stephan Hartmann *Bayesian Epistemology* by Branden Fitelson.

This essay is an attempt to gain better insight into Russell's positive account of inductive inference. I contend that Russell's postulates play only a supporting role in his overall account. At the center of Russell's positive view is a probabilistic, Bayesian model of inductive inference. Indeed, Russell and Maxwell actually held very similar Bayesian views. But the Bayesian component of Russell's view in Human Knowledge is sparse and easily overlooked. Maxwell was not aware of it when he (...) developed his own view, and I believe he was never fully aware of the extent to which Russell's account anticipates his own. The primary focus of this paper will be the explication of the Bayesian component of the Russell-Maxwell view, and the way in which it undermines judgment empiricism. (shrink)

Epistemologists often appeal to the idea that a normative theory must provide useful, usable, guidance to argue for one normative epistemology over another. I argue that this is a mistake. Guidance considerations have no role to play in theory choice in epistemology. I show how this has implications for debates about the possibility and scope of epistemic dilemmas, the legitimacy of idealisation in Bayesian epistemology, uniqueness versus permissivism, sharp versus mushy credences, and internalism versus externalism.

So-called “traditional epistemology” and “Bayesian epistemology” share a word, but it may often seem that the enterprises hardly share a subject matter. They differ in their central concepts. They differ in their main concerns. They differ in their main theoretical moves. And they often differ in their methodology. However, in the last decade or so, there have been a number of attempts to build bridges between the two epistemologies. Indeed, many would say that there is just one (...) branch of philosophy here—epistemology. There is a common subject matter after all. In this paper, we begin by playing the role of a “bad cop,” emphasizing many apparent points of disconnection, and even conflict, between the approaches to epistemology. We then switch role, playing a “good cop” who insists that the approaches are engaged in common projects after all. We look at various ways in which the gaps between them have been bridged, and we consider the prospects for bridging them further. We conclude that this is an exciting time for epistemology, as the two traditions can learn, and have started learning, from each other. (shrink)

Recently, some have challenged the idea that there are genuine norms of diachronic rationality. Part of this challenge has involved offering replacements for diachronic principles. Skeptics about diachronic rationality believe that we can provide an error theory for it by appealing to synchronic updating rules that, over time, mimic the behavior of diachronic norms. In this paper, I argue that the most promising attempts to develop this position within the Bayesian framework are unsuccessful. I sketch a new synchronic surrogate (...) that draws upon some of the features of each of these earlier attempts. At the heart of this discussion is the question of what exactly it means to say that one norm is a surrogate for another. I argue that surrogacy, in the given context, can be taken as a proxy for the degree to which formal and traditional epistemology can be made compatible. (shrink)

Jon Williamson's Objective Bayesian Epistemology relies upon a calibration norm to constrain credal probability by both quantitative and qualitative evidence. One role of the calibration norm is to ensure that evidence works to constrain a convex set of probability functions. This essay brings into focus a problem for Williamson's theory when qualitative evidence specifies non-convex constraints.

I raise skeptical doubts about the prospects of Bayesian formal epistemology for providing an adequate general normative model of epistemic rationality. The notion of credence, I argue, embodies a very dubious psychological myth, viz., that for virtually any proposition p that one can entertain and understand, one has some quantitatively precise, 0-to-1 ratio-scale, doxastic attitude toward p. The concept of credence faces further serious problems as well—different ones depending on whether credence 1 is construed as full belief (the (...) limit case of so-called partial belief) or instead is construed as absolute certainty. I argue that the notion of an “ideal Bayesian reasoner” cannot serve as a normative ideal that actual human agents should seek to emulate as closely as they can, because different such reasoners who all have the same evidence as oneself—no single one them being uniquely psychologically most similar to oneself—will differ from one another in their credences (e.g., because they commence from different prior credences). I argue that epistemic probability, properly understood, is quantitative degree of evidential support relative to one’s evidence, and that principled epistemic probabilities arise only under quite special evidential circumstances—which means that epistemic probability is ill suited to figure centrally within general norms of human epistemic rationality. (shrink)

This paper responds to Terry Horgan’s recent critique of Bayesian formal epistemology. I argue that each of Horgan’s criticisms misses its mark when Bayesianism is viewed as putting forward an inductive logic of confidences. Along the way, I explore the nature, scope, and limits of a defensible brand of Bayesianism.

Bayesian epistemology provides a popular and powerful framework for modeling rational norms on credences, including how rational agents should respond to evidence. The framework is built on the assumption that ideally rational agents have credences, or degrees of belief, that are representable by numbers that obey the axioms of probability. From there, further constraints are proposed regarding which credence assignments are rationally permissible, and how rational agents’ credences should change upon learning new evidence. While the details are hotly (...) disputed, all flavors of Bayesianism purport to give us norms of ideal rationality. This raises the question of how exactly these norms apply to you and me, since perfect compliance with those ideal norms is out of reach for human thinkers. A common response is that Bayesian norms are ideals that human reasoners are supposed to approximate – the closer they come to being ideally rational, the better. To make this claim plausible, we need to make it more precise. In what sense is it better to be closer to ideally rational, and what is an appropriate measure of such closeness? This article sketches some possible answers to these questions. (shrink)

I bring out the limitations of four important views of what the target of useful climate model assessment is. Three of these views are drawn from philosophy. They include the views of Elisabeth Lloyd and Wendy Parker, and an application of Bayesian confirmation theory. The fourth view I criticise is based on the actual practice of climate model assessment. In bringing out the limitations of these four views, I argue that an approach to climate model assessment that neither demands (...) too much of such assessment nor threatens to be unreliable will, in typical cases, have to aim at something other than the confirmation of claims about how the climate system actually is. This means, I suggest, that the Intergovernmental Panel on Climate Change ’s focus on establishing confidence in climate model explanations and predictions is misguided. So too, it means that standard epistemologies of science with pretensions to generality, e.g., Bayesian epistemologies, fail to illuminate the assessment of climate models. I go on to outline a view that neither demands too much nor threatens to be unreliable, a view according to which useful climate model assessment typically aims to show that certain climatic scenarios are real possibilities and, when the scenarios are determined to be real possibilities, partially to determine how remote they are. (shrink)

Accuracy arguments for the core tenets of Bayesian epistemology differ mainly in the conditions they place on the legitimate ways of measuring the inaccuracy of our credences. The best existing arguments rely on three conditions: Continuity, Additivity, and Strict Propriety. In this paper, I show how to strengthen the arguments based on these conditions by showing that the central mathematical theorem on which each depends goes through without assuming Additivity.