Philosophy Bayesianism
by
Kenny Easwaran
  • LAST REVIEWED: 08 October 2015
  • LAST MODIFIED: 28 May 2013
  • DOI: 10.1093/obo/9780195396577-0204

Introduction

Bayesianism is a set of related views in epistemology, statistics, philosophy of science, psychology, and any other subject that deals with notions of belief or confidence. The basic idea is that rather than being an all-or-nothing phenomenon, belief comes in degrees, and these degrees obey some formal constraints related to the axioms of probability theory. In epistemology, these views temper some traditional thoughts about belief and knowledge, and they may give rise to alternative views of justification and evidence. In philosophy of science, these views help structure views about the general practice of science. Many philosophical questions arise not only about these applications of Bayesianism, but also about the formalism used to structure it.

General Overviews

For those who are interested in learning more about a range of issues in Bayesianism, a variety of other resources are available. In particular, The Stanford Encyclopedia of Philosophy is a freely available online resource, with many relevant articles, including Hájek 2011, Hawthorne 2011, Huber 2012, Joyce 2003, and Talbott 2008. The Internet Encyclopedia of Philosophy is also quite helpful, though apart from Huber 2007, it appears to have fewer articles on these topics. Easwaran 2011a and Easwaran 2011b are also overviews that give an outline of many relevant issues, though they are not as easily available.

  • Easwaran, Kenny. “Bayesianism I: Introduction and Arguments in Favor.” Philosophy Compass 6.5 (2011a): 312–320.

    DOI: 10.1111/j.1747-9991.2011.00399.xSave Citation »Export Citation »E-mail Citation »

    Describes the views that make up Bayesianism and discusses some of the arguments that are generally used to support it. Focuses on the Dutch book and representation theorem arguments. Available online for purchase or by subscription.

    Find this resource:

    • Easwaran, Kenny. “Bayesianism II: Applications and Criticisms.” Philosophy Compass 6.5 (2011b): 321–332.

      DOI: 10.1111/j.1747-9991.2011.00398.xSave Citation »Export Citation »E-mail Citation »

      Shows some applications of Bayesianism in the philosophy of science and discusses several problems that apply there. Also gives a brief overview of some applications of Bayesianism in epistemology and in statistics. Available online for purchase or by subscription.

      Find this resource:

      • Hájek, Alan. “Interpretations of Probability.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2011.

        Save Citation »Export Citation »E-mail Citation »

        A discussion of various interpretations of probability. The logical and subjective interpretations are the most relevant for Bayesianism, but many issues that arise in the others are relevant too.

        Find this resource:

        • Hawthorne, James. “Inductive Logic.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2011.

          Save Citation »Export Citation »E-mail Citation »

          A very good discussion of some issues in inductive logic, which includes much of Bayesian confirmation theory. Has useful discussion of theorems showing that different priors will converge, given sufficiently similar evidence.

          Find this resource:

          • Huber, Franz. “Confirmation and Induction.” Internet Encyclopedia of Philosophy. 2007.

            Save Citation »Export Citation »E-mail Citation »

            Discusses several approaches to the theory of confirmation across the 20th century. Section 4 (on inductive logic) and section 6 (on Bayesian confirmation theory) are the most relevant to the topic of this bibliography.

            Find this resource:

            • Huber, Franz. “Formal Representations of Belief.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2012.

              Save Citation »Export Citation »E-mail Citation »

              Compares the probabilistic representation of belief (the basis of Bayesianism) with various other representations that have been proposed, including full belief, as well as alternative formal theories of uncertainty.

              Find this resource:

              • Joyce, James. “Bayes’ Theorem.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2003.

                Save Citation »Export Citation »E-mail Citation »

                A helpful discussion of Bayes’ theorem about conditional probabilities, showing why it is so useful in various applications in epistemology and philosophy of science.

                Find this resource:

                • Talbott, William. “Bayesian Epistemology.” In The Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2008.

                  Save Citation »Export Citation »E-mail Citation »

                  Has useful discussions of the probability axioms, Dutch book arguments, and Bayesian confirmation theory.

                  Find this resource:

                  Textbooks

                  Many textbooks are available on matters that are of relevance for Bayesianism. Skyrms 2000 and Hacking 2001 are useful introductory textbooks for students of philosophy. Savage 1972 and Jeffrey 1990 are focused more on decision theory. Halpern 2003, Jaynes and Bretthorst 2003, and Howson and Urbach 2006 are more mathematical and develop the theory in substantially different ways.

                  • Hacking, Ian. An Introduction to Probability and Inductive Logic. Cambridge, UK: Cambridge University Press, 2001.

                    DOI: 10.1017/CBO9780511801297Save Citation »Export Citation »E-mail Citation »

                    An introductory textbook aimed at students of philosophy. In addition to probability, the author introduces the ideas of decision theory and statistics with only a brief discussion of induction, despite the title.

                    Find this resource:

                    • Halpern, Joseph. Reasoning about Uncertainty. Cambridge, MA: MIT Press, 2003.

                      Save Citation »Export Citation »E-mail Citation »

                      This textbook gives a wide-ranging overview of a variety of means of reasoning about uncertainty, including many alternatives to probability theory, and also logics of belief revision.

                      Find this resource:

                      • Howson, Colin, and Peter Urbach. Scientific Reasoning: The Bayesian Approach. Chicago: Open Court, 2006.

                        Save Citation »Export Citation »E-mail Citation »

                        Although intended for a philosophical audience, this book also gives a moderately detailed introduction to various Bayesian statistical techniques and some general discussion of issues in induction.

                        Find this resource:

                        • Jaynes, Edwin T., and G. Larry Bretthorst. Probability Theory. Cambridge, UK: Cambridge University Press, 2003.

                          DOI: 10.1017/CBO9780511790423Save Citation »Export Citation »E-mail Citation »

                          This posthumous book was compiled by a student from notes that Jaynes was gathering for a large book. Philosophers will be most interested in the introductory chapters and chapter 15 on the role of infinity. The rest of the book gives a comprehensive introduction to Bayesian statistics.

                          Find this resource:

                          • Jeffrey, Richard. The Logic of Decision. Chicago: University of Chicago Press, 1990.

                            Save Citation »Export Citation »E-mail Citation »

                            Third edition of a work first published in 1965. An introduction to decision theory, including an historical overview of Ramsey’s theory and other early decision theories. However, most of the book is devoted to developing Jeffrey’s theory, introduced in the first edition.

                            Find this resource:

                            • Savage, Leonard. The Foundations of Statistics. New York: Dover, 1972.

                              Save Citation »Export Citation »E-mail Citation »

                              Originally published in 1954. The first five chapters of this book give an introduction to Savage’s version of decision theory, which he uses to lay the foundations for probability theory. The remainder of the book uses the theory so described to justify traditional statistical inference techniques, though in this later edition he accepts that more modification would be needed.

                              Find this resource:

                              • Skyrms, Brian. Choice and Chance. Belmont, CA: Wadsworth, 2000.

                                Save Citation »Export Citation »E-mail Citation »

                                An introductory textbook aimed at students of philosophy. Discusses the problem of induction as well as probability.

                                Find this resource:

                                History

                                Bayes 1763 discusses the problem of estimating a chance from some observed data, in the process proving a special case of the theorem that bears his name and after which the broader research program is since named. In the early 19th century, this theory was applied to scientific inference about the natural world by the author of Laplace 1902. After the rise of statistics in the 19th century drew attention to frequency as a type of probability, the role of probability in reasonable inference was brought back by the author of Keynes 1920, among others. Responding to Keynes’s logical conception of probability, Frank Ramsey and Bruno de Finetti independently encouraged a more private, psychological conception of it (de Finetti 1964, Ramsay 1931). Rudolf Carnap developed a very influential version that sought to return to the objectivity of Keynes’s approach, while acknowledging the relevance of another notion of probability as a physical fact (Carnap 1962). Further summaries of more detailed aspects of the history are given in Hacking 1984 for the early modern period and in von Plato 1998 for the 20th century.

                                • Bayes, Thomas. “An Essay towards Solving a Problem in the Doctrine of Chances.” Philosophical Transactions of the Royal Society of London 53 (1763): 370–418.

                                  DOI: 10.1098/rstl.1763.0053Save Citation »Export Citation »E-mail Citation »

                                  This posthumous paper gives the first proof of a version of what has become known as “Bayes’ theorem.” The paper concerns itself with estimating the chances of an outcome after observing several trials.

                                  Find this resource:

                                  • Carnap, Rudolf. The Logical Foundations of Probability. Chicago: University of Chicago Press, 1962.

                                    Save Citation »Export Citation »E-mail Citation »

                                    Second edition of a work first published in 1950. Distinguishes a physical and inferential type of probability, which Carnap interprets as frequency and logical probability, respectively. Works out theories of both, including some interesting attempts at objective priors for logical probability.

                                    Find this resource:

                                    • de Finetti, Bruno. “Foresight: Its Logical Laws, Its Subjective Sources.” In Studies in Subjective Probability. Edited by Henry Kyburg and H. E. Smokler, 93–158. Translated by Henry Kyburg. New York: Wiley, 1964.

                                      Save Citation »Export Citation »E-mail Citation »

                                      English translation of La prévision: Ses lois logique, ses sources subjectives (Paris: Institut Henri Poincaré), originally published in 1937. A strong defense of the subjective interpretation of probability. This paper introduces de Finetti’s famous results on “exchangeability,” which allows one to dispense with the notion of chance.

                                      Find this resource:

                                      • Hacking, Ian. The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction, and Statistical Inference. Cambridge, UK: Cambridge University Press, 1984.

                                        Save Citation »Export Citation »E-mail Citation »

                                        A discussion of the historical emergence of the concepts of probability in the 16th through the 18th centuries, including its connections to gambling, theology, and philosophy.

                                        Find this resource:

                                        • Keynes, John Maynard. A Treatise on Probability. London: Macmillan, 1920.

                                          Save Citation »Export Citation »E-mail Citation »

                                          A development of what would later be known as a logical theory of probability, renaming the Principle of Insufficient Reason from Laplace 1902 as the Principle of Indifference. Shows how this work can justify some techniques of statistical inference in approaching the problem of induction.

                                          Find this resource:

                                          • Laplace, Pierre Simon. A Philosophical Essay on Probabilities. Translated by Frederick Wilson Truscott and Frederick Lincoln Emory. New York: Wiley, 1902.

                                            Save Citation »Export Citation »E-mail Citation »

                                            English translation of Essai philosophique sur les probabilités (Paris: Courcier), originally published in 1814. This treatise summarizes much of Laplace’s work over the previous decades on probability theory. Chapter 3 discusses the formal theory, including the famous Principle of Insufficient Reason and the Rule of Succession (though not under those names) and applies them to various scientific problems.

                                            Find this resource:

                                            • Ramsey, Frank. “Truth and Probability.” In The Foundations of Mathematics and Other Logical Essays. Edited by R. B. Braithwaite, 159–198. Cambridge, UK: Cambridge University Press, 1931.

                                              Save Citation »Export Citation »E-mail Citation »

                                              Originally published in 1926. Develops a notion of subjective probability by means of providing the ancestor of modern representation theorems. Discusses the connections between the epistemology of full and partial belief.

                                              Find this resource:

                                              • von Plato, Jan. Creating Modern Probability Theory. Cambridge, UK: Cambridge University Press, 1998.

                                                Save Citation »Export Citation »E-mail Citation »

                                                A discussion of the development of modern measure-theoretic probability theory in the 20th century, under the influence of mathematics and physics.

                                                Find this resource:

                                                Bayesian Formalism

                                                Degrees of belief are said to obey the axioms of probability theory. One set of arguments for this view is based on the role that these degrees of belief play in motivating or justifying action. If degree of belief in a proposition is identified with the odds at which an agent is willing to bet on that proposition, then one can show that any agent whose degrees of belief don’t correspond to probabilities is vulnerable to a Dutch Books—a series of bets that she is willing to accept individually, which collectively guarantee that she loses money. Starting in the middle of the 20th century, more sophisticated versions of decision theory showed how to extract degrees of belief and utilities from an agent’s preferences over uncertain gambles. Some philosophers have preferred to separate the notion of degree of belief from any role in practical decision making; instead, they tried to argue for the probability axioms on the basis of “accuracy” as a purely epistemic notion. Probability theory comes additionally with a notion of “conditional probability.” One important question for Bayesianism has been to understand what this notion means, how it relates to the probability of ordinary language conditionals, and what role it plays in updating beliefs in light of new evidence. The main proposal has been to update through “conditionalization,” in which each proposition comes to have a new degree of belief equal to its old probability conditional on the evidence that is learned, but the alternatives involve conditional probability as well. Many worries about these formalisms become particularly acute when the set of possibilities over which the agent is uncertain is infinite, so that many possibilities have probability 0.

                                                Dutch Books

                                                The traditional argument that degree of belief ought to obey the axioms of probability theory involves showing that if degrees of belief measure the price that an agent considers fair for either buying or selling a bet on a proposition, then an agent whose degrees of belief don’t obey the probability axioms has a set of bets that she finds individually fair but that collectively guarantee a loss for her. Such a set of bets is called a “Dutch book.” Because such a collection of bets is clearly unfair, it is argued that any rational agent must have degrees of belief that satisfy the probability axioms and, perhaps, further constraints depending on the specific argument. For instance, Teller 1973 argues that a rational Bayesian agent must update her degrees of belief by the rule of conditionalization. This argument is based on work by David Lewis that was eventually published as Lewis 1999. A common worry about Dutch book arguments is that they seem to show that violation of the probability axioms leads only to a monetary loss, which seems different from an epistemic failing. Christensen 1996 and Skyrms 1992 argue that the proper way to interpret the arguments is to demonstrate the inconsistency of such an agent’s evaluations, and the inconsistency is itself the problem, not the monetary loss. Christensen 1991 argues that “diachronic” Dutch books, in which the bets are offered at different times, fail to show incoherence in this sense and, thus, can’t be used to establish requirements of rationality. McGee 1999 demonstrates an interesting Dutch book that seems to show that it is irrational to allow both arbitrarily high stakes on bets and arbitrarily low probabilities of outcomes. However, this “airtight” Dutch book may, instead, demonstrate a weakness in the type of argument.

                                                • Christensen, David. “Clever Bookies and Coherent Beliefs.” Philosophical Review 100.2 (1991): 229–247.

                                                  DOI: 10.2307/2185301Save Citation »Export Citation »E-mail Citation »

                                                  Argues that diachronic Dutch book arguments can’t establish requirements of rationality. He does this by comparing diachronic coherence to interpersonal coherence and arguing that there is no rational requirement for either. Available online by subscription.

                                                  Find this resource:

                                                  • Christensen, David. “Dutch-Book Arguments Depragmatized: Epistemic Consistency for Partial Believers.” Journal of Philosophy 93.9 (1996): 450–479.

                                                    DOI: 10.2307/2940893Save Citation »Export Citation »E-mail Citation »

                                                    Suggests that the proper way to interpret Dutch book arguments can remove their pragmatic elements. Degrees of belief give rational requirements on value, and values must be consistent. It is inconsistency among values that show irrationality, not actual loss or gain of money.

                                                    Find this resource:

                                                    • Lewis, David. “Why Conditionalize?” In Papers in Metaphysics and Epistemology. Vol. 2. Edited by David Lewis, 403–407. Cambridge, UK: Cambridge University Press, 1999.

                                                      DOI: 10.1017/CBO9780511625343.024Save Citation »Export Citation »E-mail Citation »

                                                      Much later publication of some notes from 1972 that form the basis of the argument in Teller 1973. Gives both a synchronic and a diachronic argument to connect unconditional probabilities to conditional probabilities as well as update rules.

                                                      Find this resource:

                                                      • McGee, Vann. “An Airtight Dutch Book.” Analysis 59.264 (1999): 257–265.

                                                        DOI: 10.1093/analys/59.4.257Save Citation »Export Citation »E-mail Citation »

                                                        Demonstrates a “double-or-nothing” type of strategy that can be repeated over and over to guarantee loss for an agent with coherent credences, arbitrarily small probabilities, and arbitrarily large utilities. Uses this to suggest some problems for Dutch book arguments. Available online for purchase or by subscription.

                                                        Find this resource:

                                                        • Skyrms, Brian. “Coherence, Probability, and Induction.” Philosophical Issues 2 (1992): 215–226.

                                                          DOI: 10.2307/1522864Save Citation »Export Citation »E-mail Citation »

                                                          Suggests several ways that Dutch book arguments should be modified to properly account for various worries. In addition to “depragmatizing” them, Skyrms also suggests considering only favorable (rather than fair) bets and allowing bets on one’s future bets, and not just first-order events. Available online by subscription.

                                                          Find this resource:

                                                          • Teller, Paul. “Conditionalization and Observation.” Synthese 26.2 (1973): 218–258.

                                                            DOI: 10.1007/BF00873264Save Citation »Export Citation »E-mail Citation »

                                                            In section 1.3, Teller describes a Dutch book argument he attributes to David Lewis, which shows that rational updates ought to obey conditionalization. This is one of the first diachronic Dutch book arguments. Available online for purchase or by subscription.

                                                            Find this resource:

                                                            Bayesian Decision Theory

                                                            Ramsey 1931 introduces the idea that an agent’s degrees of belief can be read off from her disposition to accept or reject bets on various propositions, and a numerical scale of values for the agent could be constructed in the same way. Savage 1972 develops a much more sophisticated version of this argument, based on a general notion of “actions” as functions from “states” to “outcomes.” The result is a “representation theorem” showing that any agent whose preferences satisfy certain structural axioms must be represented by a unique probability function and a utility function that is unique up to affine transformation. Jeffrey 1990 modifies this argument so as not to draw a significant metaphysical distinction between the propositions that are considered actions of the agent and those that are considered to be part of the world, and, as a result, his representation theorem no longer gives precise uniqueness to the representation. Jeffrey’s framework introduced a further complication involving the possibility of features of the world that are correlated with the agent’s actions in ways that intuitively seem irrelevant for decision making. A useful summary of the literature responding to this complication is Joyce 1999. Questions about the relevance of the representation theorems have been raised in Zynda 2000 and Meacham and Weisberg 2011. But Dreier 1996 argues that representation theorems are essential to a natural understanding of rational decision making. Many have objected to the representation theorems as requiring an unrealistic neutrality to risk for all rational agents. The extensive literature on this issue is summarized and material is added in Buchak 2009.

                                                            • Buchak, Lara. “Risk and Rationality.” PhD diss., Princeton University, 2009.

                                                              Save Citation »Export Citation »E-mail Citation »

                                                              Weakens the “sure-thing principle” in Savage 1972 to allow that agents might be sensitive to risk in their decision making. Proves a representation theorem for the weaker axioms, allowing a parameter to measure this risk sensitivity.

                                                              Find this resource:

                                                              • Dreier, James. “Rational Preference: Decision Theory as a Theory of Practical Rationality.” Theory and Decision 40 (1996): 249–276.

                                                                DOI: 10.1007/BF00134210Save Citation »Export Citation »E-mail Citation »

                                                                Argues that there is no useful interpretation of the notions of degree of belief or subjective utility apart from their role in decision making. Argues that the Humean picture of rationality is best understood by means of representation theorems. Available online for purchase or by subscription.

                                                                Find this resource:

                                                                • Jeffrey, Richard. The Logic of Decision. Chicago: University of Chicago Press, 1990.

                                                                  Save Citation »Export Citation »E-mail Citation »

                                                                  Includes textbook summaries of Ramsey 1931 and Savage 1972. Develops an alternate theory that puts all propositions on a par, whether they involve the agent’s actions or not. Shows that utility must be bounded and that the width of the bounds is related to the precision with which probabilities are determined.

                                                                  Find this resource:

                                                                  • Joyce, James. The Foundations of Causal Decision Theory. Cambridge, UK: Cambridge University Press, 1999.

                                                                    DOI: 10.1017/CBO9780511498497Save Citation »Export Citation »E-mail Citation »

                                                                    Develops versions of the decision theory of Jeffrey 1990 and shows how to modify it to account for the difference between “news value” and “causal value” of an action. Discusses some alternative notions of conditional probability to help with this.

                                                                    Find this resource:

                                                                    • Meacham, Christopher, and Jonathan Weisberg. “Representation Theorems and the Foundations of Decision Theory.” Australasian Journal of Philosophy 89 (2011): 641–663.

                                                                      DOI: 10.1080/00048402.2010.510529Save Citation »Export Citation »E-mail Citation »

                                                                      Considers empirical and nonempirical understandings of the representation theorems. Argues that empirical understandings don’t have enough empirical justification, and nonempirical understandings stipulate notion of belief and desire that have no connection with our actual attitudes.

                                                                      Find this resource:

                                                                      • Ramsey, Frank. “Truth and Probability.” In The Foundations of Mathematics and Other Logical Essays. Edited by R. B. Braithwaite, 159–198. Cambridge, UK: Cambridge University Press, 1931.

                                                                        Save Citation »Export Citation »E-mail Citation »

                                                                        One of the first considerations of degrees of belief. Argues that the proper way to measure strength is not an internal phenomenological one but rather by the role it plays in decision making. Shows how to measure both belief and utility numerically on this basis.

                                                                        Find this resource:

                                                                        • Savage, Leonard. The Foundations of Statistics. New York: Dover, 1972.

                                                                          Save Citation »Export Citation »E-mail Citation »

                                                                          The first five chapters develop a theory of probability and utility out of rationality constraints on decision making, focusing on “the sure-thing principle.” The resulting theory is orthodoxy among many decision theorists outside of philosophy.

                                                                          Find this resource:

                                                                          • Zynda, Lyle. “Representation Theorems and Realism about Degrees of Belief.” Philosophy of Science 67 (2000): 45–69.

                                                                            DOI: 10.1086/392761Save Citation »Export Citation »E-mail Citation »

                                                                            Considers views called anti-realism, weak realism, and strong realism about degrees of belief. Argues that with a strong realist view, representation theorems are insufficient to show that degrees of belief ought to obey the probability axioms.

                                                                            Find this resource:

                                                                            Accuracy Arguments

                                                                            Accuracy is a notion of “closeness to truth” that some believe to be a properly epistemic consideration to replace the merely pragmatic considerations of Dutch book arguments and decision theory. Joyce 1998 is perhaps the first to put the argument in these terms. This argument is strengthened and extended in Joyce 2009. This argument is based on a reinterpretation of the “scoring rules” introduced in de Finetti 1974 for a similar purpose and generalized in Lindley 1982. However, de Finetti 1974 and Lindley 1982 interpret the scoring rule as some sort of pragmatic penalty rather than as a purely epistemic value. Leitgeb and Pettigrew 2010a and Leitgeb and Pettigrew 2010b give a generalization of Joyce’s argument to establish not just probabilism, but further principles as well. These arguments are summarized and compared to others in Pettigrew 2011.

                                                                            Conditional Probability and Probability of Conditionals

                                                                            The traditional notion of the conditional probability of one event A given another event B is defined in Kolmogorov 1956 as P(A|B) = P(A&B)/P(B). This notion plays an important role in many statistical tests and rules for responding to evidence. Adams 1965 proposes that it could also be used to help understand the indicative conditional of ordinary language, proposing that the probability of a conditional is the corresponding conditional probability. Lewis 1976 shows that this claim can be true only if the probability function is “trivial,” in the sense that it only takes on two values. Edgington 1995 develops the Adams theory in light of the Lewis results by taking conditionals not to have truth values. More recently, the author of Hajek 2003 has argued that conditional probability should not be defined in terms of unconditional probability; rather, it must be understood as an independent (and perhaps more fundamental) notion, perhaps along the lines suggested in Popper 1955 or Rényi 1955. Easwaran 2008 suggests that Popper’s axioms fail to respect important features of probability and that, instead, Kolmogorov’s more sophisticated version should be used.

                                                                            • Adams, Ernest. “The Logic of Conditionals.” Inquiry 8 (1965): 166–197.

                                                                              DOI: 10.1080/00201746508601430Save Citation »Export Citation »E-mail Citation »

                                                                              Argues that the probability of a conditional should be understood as the corresponding conditional probability. Develops a logic that gives a notion of entailment avoiding many paradoxes of classical entailment for conditionals. Available online for purchase or by subscription.

                                                                              Find this resource:

                                                                              • Easwaran, Kenny. “The Foundations of Conditional Probability.” PhD diss., University of California at Berkeley, 2008.

                                                                                Save Citation »Export Citation »E-mail Citation »

                                                                                Considers the role of conditional probability in scientific evidence to argue for a principle known as “conglomerability”—if some members of a partition give conditional probabilities for A that are lower than its unconditional probability, then some other members must give it higher conditional probability; no experiment can be guaranteed to provide evidence against A.

                                                                                Find this resource:

                                                                                • Edgington, Dorothy. “On Conditionals.” Mind 104.414 (1995): 235–329.

                                                                                  DOI: 10.1093/mind/104.414.235Save Citation »Export Citation »E-mail Citation »

                                                                                  Gives an extensive summary of the literature on probability and conditionals. Develops various versions of the account that get around the triviality results. Connects this with a theory of counterfactual conditionals as well. Available online for purchase or by subscription.

                                                                                  Find this resource:

                                                                                  • Hájek, Alan. “What Conditional Probability Could Not Be.” Synthese 137 (2003): 273–323.

                                                                                    DOI: 10.1023/B:SYNT.0000004904.91112.16Save Citation »Export Citation »E-mail Citation »

                                                                                    By considering infinite probability spaces, and ones in which various events have no well-defined probability, argues that many conditional probabilities can still be understood, and thus can’t be defined in terms of unconditional probability as proposed by Kolmogorov.

                                                                                    Find this resource:

                                                                                    • Kolmogorov, A. N. Foundations of the Theory of Probability. Edited and translated by Nathan Morrison. New York: Chelsea, 1956.

                                                                                      Save Citation »Export Citation »E-mail Citation »

                                                                                      English translation of Grundbegriffe der Wahrscheinlichkeitsrechnung (Berlin: J. Springer), originally published in 1933. Gives the first modern axiomatic characterization of probability theory. In particular, although chapter 1 gives the standard definition of conditional probability, chapter 5 defines an extension to infinite spaces that avoids some problems.

                                                                                      Find this resource:

                                                                                      • Lewis, David. “Probabilities of Conditionals and Conditional Probabilities.” Philosophical Review 85.3 (1976): 297–315.

                                                                                        DOI: 10.2307/2184045Save Citation »Export Citation »E-mail Citation »

                                                                                        Shows that it is impossible for the probability of a conditional to always equal the corresponding conditional probability, except for trivial languages or trivial probability functions. Suggests that, instead, we should think of conditionals as having no truth values and, thus, not being propositions that have probabilities. Available online by subscription.

                                                                                        Find this resource:

                                                                                        • Popper, Karl. “Two Autonomous Axiom Systems for the Calculus of Probabilities.” British Journal for the Philosophy of Science 6 (1955): 51–57.

                                                                                          DOI: 10.1093/bjps/VI.21.51Save Citation »Export Citation »E-mail Citation »

                                                                                          Gives an alternate axiomatization of probability theory in terms of conditional probability, which allows every conditional probability to be defined, regardless of which unconditional probabilities are 0. Shows that this axiomatization characterizes classical logic along with probability, instead of presupposing it. Available online by subscription.

                                                                                          Find this resource:

                                                                                          • Rényi, Alfréd. “On a New Axiomatic Theory for Probability.” Acta Mathematica Hungarica 6 (1955): 285–335.

                                                                                            DOI: 10.1007/BF02024393Save Citation »Export Citation »E-mail Citation »

                                                                                            Gives an alternate axiomatization of probability theory in terms of conditional probability. Some philosophers take this axiomatization to help with the problem of probability zero, but it is intended to deal with the converse problem, of mathematical situations that motivate unboundedly large probabilities. Available online for purchase or by subscription.

                                                                                            Find this resource:

                                                                                            Conditionalization and Its Alternatives

                                                                                            In addition to the basic probability axioms, Bayesians argue that an agent ought to update her degrees of belief by a rule known as “conditionalization.” That is, when an agent learns a piece of evidence E, she must come to have a new degree of belief in A given by her old P(A|E). I. J. Good shows that this update rule satisfies the nice property that an agent should never turn down free information—the agent will always expect to do better by updating in this way before making a decision. Teller 1973 gives an early set of arguments in favor of this rule. Greaves and Wallace 2006 gives a more modern argument that uses a notion of “epistemic utility.” Van Fraassen 1999 gives one that uses the author’s “reflection principle.” Jeffrey 1990 argues that this rule was too restricted—in many cases (or perhaps even in all cases) the agent fails to become certain of her evidence but just increases her confidence in it. Jeffrey proposed that, in this sort of case, an agent should have Pnew(A) = Pold(A|E)Pnew(E) + Pold(A|-E)Pnew(-E). This rule reduces to conditionalization in the case where Pnew(E) = 1. Myrvold 2010 generalizes the result in Good 1967 to apply to this rule as well as to generalizations of it. Leitgeb and Pettigrew 2010 proposes a competitor to Jeffrey’s rule for these same cases.

                                                                                            • Good, I. J. “On the Principle of Total Evidence.” British Journal for the Philosophy of Science 17 (1967): 319–321.

                                                                                              DOI: 10.1093/bjps/17.4.319Save Citation »Export Citation »E-mail Citation »

                                                                                              Justifies Carnap’s principle of total evidence by showing that if an agent plans to first gather some free information, update by conditionalization, and then choose an action of maximum expected value, then the agent expects to do better overall than if she just chooses the act that maximizes expected value now.

                                                                                              Find this resource:

                                                                                              • Greaves, Hilary, and David Wallace. “Justifying Conditionalization: Conditionalization Maximizes Expected Epistemic Utility.” Mind 115.459 (2006): 607–632.

                                                                                                DOI: 10.1093/mind/fzl607Save Citation »Export Citation »E-mail Citation »

                                                                                                Considers an update situation as given by a partition of potential pieces of evidence an agent might learn. Shows that if the agent makes an update plan that maximizes expected epistemic utility, then the agent must plan to update by conditionalization. Available online for purchase or by subscription.

                                                                                                Find this resource:

                                                                                                • Jeffrey, Richard. The Logic of Decision. Chicago: University of Chicago Press, 1990.

                                                                                                  Save Citation »Export Citation »E-mail Citation »

                                                                                                  Chapter 11 of this book discusses what Jeffrey calls “probability kinematics,” which is his alternative to conditionalization. He argues that we can never be fully certain of anything and, thus, we ought to update by means of this method.

                                                                                                  Find this resource:

                                                                                                  • Leitgeb, Hannes, and Richard Pettigrew. “An Objective Justification of Bayesianism II: The Consequences of Minimizing Inaccuracy.” Philosophy of Science 77.2 (2010): 236–272.

                                                                                                    DOI: 10.1086/651318Save Citation »Export Citation »E-mail Citation »

                                                                                                    Among other things, develops an alternative to Jeffrey conditionalization that the authors argue is superior on accuracy grounds. This alternative is complicated and implausible, but it is one of the few challengers to Jeffrey conditionalization in the situations it is intended for.

                                                                                                    Find this resource:

                                                                                                    • Myrvold, Wayne. “Epistemic Values and the Value of Learning.” Synthese 187.2 (2010): 547–568.

                                                                                                      DOI: 10.1007/s11229-010-9860-xSave Citation »Export Citation »E-mail Citation »

                                                                                                      Shows that if an agent updates by any method that satisfies a general stability condition (which both standard and Jeffrey conditionalization satisfy), then she must expect her accuracy to go up over the update. The result found in Good 1967 on the value of learning is a corollary. Available online for purchase or by subscription.

                                                                                                      Find this resource:

                                                                                                      • Teller, Paul. “Conditionalization and Observation.” Synthese 26.2 (1973): 218–258.

                                                                                                        DOI: 10.1007/BF00873264Save Citation »Export Citation »E-mail Citation »

                                                                                                        Teller considers several types of argument (frequency, Dutch book, and a sort of stability) and shows that they all support conditionalization. In the second half of the article, he shows that these arguments support Jeffrey conditionalization as well when applied properly. Available online.

                                                                                                        Find this resource:

                                                                                                        • van Fraassen, Bas. “Conditionalization, a New Argument For.” Topoi 18.2 (1999): 93–96.

                                                                                                          DOI: 10.1023/A:1006286003463Save Citation »Export Citation »E-mail Citation »

                                                                                                          Shows that if an agent updates in a way that respects a reflection principle (that the expected value of any random variable for her now is within the range spanned by her possible expected values for that random variable after updating), then the agent must update by conditionalization. Available online for purchase or by subscription.

                                                                                                          Find this resource:

                                                                                                          Infinity and Zero

                                                                                                          In the standard treatment of probability theory, special issues arise around infinity and zero. The most basic axioms of probability theory say that if two events are incompatible, then the probability that one of them happens is the sum of the probabilities of each. But in modern probability theory, this additivity is generalized to countably infinite collections of events. Seidenfeld and Schervish 1983 shows that the denial of this countable additivity in de Finetti 1974 (see section on Accuracy Arguments) is in tension with de Finetti’s use of Dutch book arguments, and Williamson 1999 gives a more direct proof of this result. However, Arntzenius, et al. 2004 shows that the addition of infinitely many bets can cause other problems, undermining these arguments. Seidenfeld, et al. 1986 shows that restricting to mere finite additivity causes other problems for certain methods in statistics. When there are many possible situations that are uncountable, then even finite additivity suffices to show that most of them have probability zero. Hájek 2003 shows that this causes problems for the traditional notion of conditional probability, and Easwaran 2008b deals with these problems in a different way. Nover and Hájek 2004 also shows that if there are infinitely many possible situations, then even some individual actions may fail to have a well-defined value. Easwaran 2008a shows one way to assign values to some such actions.

                                                                                                          • Arntzenius, Frank, Adam Elga, and John Hawthorne. “Bayesianism, Infinite Decisions, and Binding.” Mind 113.450 (2004): 251–283.

                                                                                                            DOI: 10.1093/mind/113.450.251Save Citation »Export Citation »E-mail Citation »

                                                                                                            Discusses six puzzles involving infinitely many decisions. Argues that the response to any of them must deny the validity of infinite Dutch book arguments. Suggests that diachronic and synchronic versions of the puzzles will work out quite differently from one another. Available online for purchase or by subscription.

                                                                                                            Find this resource:

                                                                                                            • Easwaran, Kenny. “Strong and Weak Expectations.” Mind 117.467 (2008a): 633–641.

                                                                                                              DOI: 10.1093/mind/fzn053Save Citation »Export Citation »E-mail Citation »

                                                                                                              Considers the “Pasadena game” of Nover and Hájek 2004, and demonstrates a notion of “weak expectation” that applies to it. Allows that some more complicated games may nevertheless fail to have any sort of expected value. Available online for purchase or by subscription.

                                                                                                              Find this resource:

                                                                                                              • Easwaran, Kenny. “The Foundations of Conditional Probability.” PhD diss., University of California at Berkeley, 2008b.

                                                                                                                Save Citation »Export Citation »E-mail Citation »

                                                                                                                Argues that conditional probabilities should be conglomerable even in infinitary situations, which restricts the way conditional probability can work for events of probability zero. In the last chapter, also considers an alternative in which events with probability zero are impossible.

                                                                                                                Find this resource:

                                                                                                                • Hájek, Alan. “What Conditional Probability Could Not Be.” Synthese 137 (2003): 273–323.

                                                                                                                  DOI: 10.1023/B:SYNT.0000004904.91112.16Save Citation »Export Citation »E-mail Citation »

                                                                                                                  Shows that when there are many uncountable possibilities, most of them must have probability zero. Shows that this causes problems for the traditional understanding of conditional probability, which must then be understood to be primitive.

                                                                                                                  Find this resource:

                                                                                                                  • Nover, Harris, and Alan Hájek. “Vexing Expectations.” Mind 113.450 (2004): 305–317.

                                                                                                                    Save Citation »Export Citation »E-mail Citation »

                                                                                                                    Defines a game called the “Pasadena game,” which has no well-defined expected value. This game has much in common with the classic “St. Petersburg game” of Bernoulli, which has infinite expected value. Available online for purchase or by subscription.

                                                                                                                    Find this resource:

                                                                                                                    • Seidenfeld, Teddy, and Mark Schervish. “A Conflict between Finite Additivity and Avoiding Dutch Book.” Philosophy of Science 50.3 (1983): 398–412.

                                                                                                                      DOI: 10.1086/289126Save Citation »Export Citation »E-mail Citation »

                                                                                                                      Shows in the framework of Savage’s decision theory that, under a particular technical assumption about the existence of events with a guaranteed outcome, there is a Dutch book against agents who violate countable additivity. Suggests some ways to resolve the tension. Available online by subscription.

                                                                                                                      Find this resource:

                                                                                                                      • Seidenfeld, Teddy, Mark Schervish, and Joseph Kadane. “Statistical Implications of Finitely Additive Probability.” In Bayesian Inference and Decision Techniques. Edited by Prem Goel and Arnold Zellner, 59–76. Amsterdam: Elsevier, 1986.

                                                                                                                        Save Citation »Export Citation »E-mail Citation »

                                                                                                                        Shows that finitely additive probabilities fail to be conglomerable. Shows that this undermines the use of certain dominance rules. Suggests that the statistical technique of “improper priors” mandates merely finitely additive probability, so that these dominance rules should be given up.

                                                                                                                        Find this resource:

                                                                                                                        • Williamson, Jon. “Countable Additivity and Subjective Probability.” British Journal for the Philosophy of Science 50.3 (1999): 401–416.

                                                                                                                          DOI: 10.1093/bjps/50.3.401Save Citation »Export Citation »E-mail Citation »

                                                                                                                          Discusses the objections to countable additivity in de Finetti 1974 (cited under Accuracy Arguments), and shows that violations of countable additivity lead to a Dutch book. Available online for purchase or by subscription.

                                                                                                                          Find this resource:

                                                                                                                          Bayesian Epistemology

                                                                                                                          As a theory of belief and its justification, Bayesianism naturally relates to many topics in epistemology. One topic that has received considerable interest recently is that of “self-locating belief,” which is belief not about which world is actual but about which person in that world one happens to be. But probability has also been used to analyze other issues from epistemology. Roush 2005 uses probabilities to develop a theory of knowledge that avoids the problems of traditional ones. Bovens and Hartmann 2004 applies Bayesian techniques to clarify several notions that arise in existing theories of knowledge. Weatherson 2007 and Kotzen 2012 both consider some traditional epistemological responses to the problem of skepticism and show how they fare on a Bayesian picture. Another important topic throughout the history of Bayesianism is the way one ought to form one’s degrees of belief. One set of approaches has been various principles of “direct inference,” in which one adopts some external probability function as one’s degrees of belief. A more general approach is that of “objective Bayesianism,” which claims that, in any evidential situation, there is one objectively correct set of degrees of belief that one ought to have. There are far more topics on Bayesianism and epistemology than can be covered here, but some others are addressed in the article on Formal Epistemology.

                                                                                                                          • Bovens, Luc, and Stephan Hartmann. Bayesian Epistemology. Oxford: Oxford University Press, 2004.

                                                                                                                            DOI: 10.1093/0199269750.001.0001Save Citation »Export Citation »E-mail Citation »

                                                                                                                            Applies Bayesian techniques to a variety of issues in epistemology. One chapter each is devoted to information, coherence, reliabilism, confirmation, and testimony.

                                                                                                                            Find this resource:

                                                                                                                            • Kotzen, Matthew. “Dragging and Confirming.” Philosophical Review 121 (2012): 55–93.

                                                                                                                              DOI: 10.1215/00318108-1426364Save Citation »Export Citation »E-mail Citation »

                                                                                                                              Considers the putative condition that if evidence confirms a proposition, then it confirms all consequences of that proposition. After showing that it is false, this paper gives conditions under which it nevertheless holds and shows that these conditions can help understand issues like warrant transmission in epistemology. Available online for purchase or by subscription.

                                                                                                                              Find this resource:

                                                                                                                              • Roush, Sherrilyn. Tracking Truth. Oxford: Oxford University Press, 2005.

                                                                                                                                DOI: 10.1093/0199274738.001.0001Save Citation »Export Citation »E-mail Citation »

                                                                                                                                Develops a version of the “tracking” theory of knowledge that uses probabilities in place of counterfactual conditionals. Shows that this version of the theory can avoid the major problems of the traditional version.

                                                                                                                                Find this resource:

                                                                                                                                • Weatherson, Brian. “The Bayesian and the Dogmatist.” Proceedings of the Aristotelian Society 107 (2007): 169–185.

                                                                                                                                  DOI: 10.1111/j.1467-9264.2007.00217.xSave Citation »Export Citation »E-mail Citation »

                                                                                                                                  Considers the “dogmatist” response to skepticism suggested by James Pryor and shows that adopting it requires a modification of the Bayesian picture. In particular, Weatherson suggests that we should represent agents by a set of probability functions and update by eliminating functions as well as conditionalizing. Available online for purchase or by subscription.

                                                                                                                                  Find this resource:

                                                                                                                                  Self-Locating Belief

                                                                                                                                  A prominent topic of recent research is the question of how information purely about what the world is like can relate to information purely about one’s location in the world. The distinction was introduced by several philosophers around 1980, but that drawn by David Lewis is the one that is most often discussed in the Bayesian literature (Lewis 1979). This issue has been brought to the attention of Bayesians by Adam Elga, who popularized the “Sleeping Beauty” thought experiment (Elga 2000). In this experiment, the agent knows that if a coin flip came up heads then today must be Monday, while if it came up tails then she will have exactly these same experiences on both Monday and Tuesday. The “thirder” position, defended in Elga 2000 and Titelbaum 2008, claims that because the agent’s evidence can be achieved two ways in tails worlds and only one way in heads worlds, it should result in the agent having a degree of belief ⅓ in heads. The “halfer” position, defended in Meacham 2008, Bradley 2012, and Bostrom 2007, claims that only the chances are relevant, so the degree of belief in heads should be ½. Among halfers, a further dispute arises about whether learning that today is Monday should affect the degrees of belief further. Although the puzzle case is very artificial, Arntzenius 2003 shows that similar issues arise in a variety of other cases. These issues are perhaps most relevant in discussions of the Anthropic Principle and related issues in physics.

                                                                                                                                  • Arntzenius, Frank. “Some Problems for Conditionalization and Reflection.” Journal of Philosophy 100.7 (2003): 356–370.

                                                                                                                                    Save Citation »Export Citation »E-mail Citation »

                                                                                                                                    Considers five example situations, one of which is Sleeping Beauty. Argues that the rational responses to these cases involve changes in degree of belief that violate standard principles of conditionalization and reflection, partly because of the phenomenon of self-location. Available online by subscription.

                                                                                                                                    Find this resource:

                                                                                                                                    • Bostrom, Nick. “Sleeping Beauty and Self-Location: A Hybrid Model.” Synthese 157.1 (2007): 59–78.

                                                                                                                                      DOI: 10.1007/s11229-006-9010-7Save Citation »Export Citation »E-mail Citation »

                                                                                                                                      Considers the Sleeping Beauty problem and argues for the halfer solution. Through consideration of various related cases (some of which are directly relevant for anthropic reasoning), the author argues that learning that today is Monday should be irrelevant. Available online for purchase or by subscription.

                                                                                                                                      Find this resource:

                                                                                                                                      • Bradley, Darren. “Four Problems about Self-Locating Belief.” Philosophical Review 121.2 (2012): 149–177.

                                                                                                                                        DOI: 10.1215/00318108-1539071Save Citation »Export Citation »E-mail Citation »

                                                                                                                                        Considers four types of situation, including Sleeping Beauty and some anthropic situations, and argues for the halfer position, showing how it is relevant to views on other issues. Uses evidence selection effects to make the arguments. Available online for purchase or by subscription.

                                                                                                                                        Find this resource:

                                                                                                                                        • Elga, Adam. “Self-Locating Belief and the Sleeping Beauty Problem.” Analysis 60.266 (2000): 143–147.

                                                                                                                                          DOI: 10.1093/analys/60.2.143Save Citation »Export Citation »E-mail Citation »

                                                                                                                                          Introduces the Sleeping Beauty problem to the philosophical literature and draws attention to Lewis’s early work on self-location. Argues for the thirder position. Available online for purchase or by subscription.

                                                                                                                                          Find this resource:

                                                                                                                                          • Lewis, David. “Attitudes De Dicto and De Se.” Philosophical Review 88.4 (1979): 513–543.

                                                                                                                                            DOI: 10.2307/2184843Save Citation »Export Citation »E-mail Citation »

                                                                                                                                            Considers the traditional distinction between de re and de dicto attitudes and suggests that both are really special cases of de se, where one has an attitude to the content that one is at a particular location in a particular world rather than just a content about a particular object or what the world is like. Available online by subscription.

                                                                                                                                            Find this resource:

                                                                                                                                            • Meacham, Christopher. “Sleeping Beauty and the Dynames of De Se Belief.” Philosophical Studies 138.2 (2008): 245–269.

                                                                                                                                              DOI: 10.1007/s11098-006-9036-1Save Citation »Export Citation »E-mail Citation »

                                                                                                                                              Considers cases analogous to the Sleeping Beauty case and investigates what the thirder position and both halfer positions say about them. Suggests that all give implausible responses to the relevant cases, but that one of the halfer positions gives the least implausible response. Develops a corresponding theory of belief update for self-locating beliefs. Available online for purchase or by subscription.

                                                                                                                                              Find this resource:

                                                                                                                                              • Titelbaum, Michael. “The Relevance of Self-Locating Beliefs.” Philosophical Review 117.4 (2008): 555–606.

                                                                                                                                                DOI: 10.1215/00318108-2008-016Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                Develops a framework for updating degrees of belief in situations involving self-locating and non-self-locating propositions. Shows that this framework gives the intuitively correct answer for many simple cases that other models fail to give as well as the thirder solution for Sleeping Beauty. Available online for purchase or by subscription.

                                                                                                                                                Find this resource:

                                                                                                                                                Direct Inference

                                                                                                                                                One of the most natural suggestions for how to set one’s degrees of belief in an event is to say that (in the absence of countervailing information) they should equal the chances or frequencies of the event type, when one of these is known. Principles giving such connections are known as principles of “direct inference.” Miller 1966 proposes a paradox for early versions of these principles, but Good 1970 shows how to avoid this particular worry. Levi 1977 argues that direct inference principles can help us understand the relations between chance and frequency, though they must be carefully constrained. Lewis 1980 shows how a version of direct inference that the author called the “Principal Principle” can help clarify the metaphysics of chance, though it caused problems for his preferred theory. Hall 1994 gives an important revision to this principle that helps save Lewis’s preferred theory of chance, but Briggs 2009 shows that there are still further problems. Pollock 1990 takes a very different approach to the problem, which has made it hard to integrate this author’s work into this particular literature.

                                                                                                                                                • Briggs, Rachael. “The Anatomy of the Big Bad Bug.” Noûs 43 (2009): 428–449.

                                                                                                                                                  DOI: 10.1111/j.1468-0068.2009.00713.xSave Citation »Export Citation »E-mail Citation »

                                                                                                                                                  Shows that the attempt to fix the Principal Principle in Hall 1994 together with many others either restrict it too far or cause further problems. Argues that this should motivate a rejection of Lewis’s views on chance. Available online for purchase or by subscription.

                                                                                                                                                  Find this resource:

                                                                                                                                                  • Good, I. J. “A Suggested Resolution of Miller’s Paradox.” British Journal for the Philosophy of Science 21 (1970): 288–289.

                                                                                                                                                    DOI: 10.1093/bjps/21.3.288Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                    Shows that the paradox in Miller 1966 turns on a distinction between terms that refer to a known numerical value and ones that refer to an unknown value. Suggests that direct inference principles can be saved by careful attention to this distinction. Available online for purchase or by subscription.

                                                                                                                                                    Find this resource:

                                                                                                                                                    • Hall, Ned. “Correcting the Guide to Objective Chance.” Mind 103.412 (1994): 505–518.

                                                                                                                                                      DOI: 10.1093/mind/103.412.505Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                      Addresses the problem in Lewis 1980 of “self-undermining” chance functions by conditioning each chance function on the claim that it is the correct function. This “New Principle” is compatible with Lewis’s metaphysical views on chance, but it has spawned a literature considering other fixes. Available online by subscription.

                                                                                                                                                      Find this resource:

                                                                                                                                                      • Levi, Isaac. “Direct Inference.” Journal of Philosophy 74 (1977): 5–29.

                                                                                                                                                        DOI: 10.2307/2025732Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                        Considers various proposals of direct inference principles and shows how they relate to chance and frequency. Shows that ones based on frequency run into problems with conditionalization while ones based on chance require a more robust understanding of chance than many prefer. Available online by subscription.

                                                                                                                                                        Find this resource:

                                                                                                                                                        • Lewis, David. “A Subjectivist’s Guide to Objective Chance.” In Studies in Inductive Logic and Probability. Vol. 2. Edited by Richard Jeffrey, 263–293. Berkeley: University of California Press, 1980.

                                                                                                                                                          Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                          Argues for a particular connection, called the “Principal Principle,” between chance and degree of belief. Shows how this principle can be used to understand chance given an understanding of degrees of belief. Shows also that it conflicts with Lewis’s “Humean” view of chance.

                                                                                                                                                          Find this resource:

                                                                                                                                                          • Miller, David. “A Paradox of Information.” British Journal for the Philosophy of Science 17 (1966): 59–61.

                                                                                                                                                            Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                            Demonstrates an apparent contradiction that can be derived from any sort of direct inference principle, whether based on chances, frequencies, or any other probability function. Available online for purchase or by subscription.

                                                                                                                                                            Find this resource:

                                                                                                                                                            • Pollock, John. Nomic Probability and the Foundations of Induction. Oxford: Oxford University Press, 1990.

                                                                                                                                                              Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                              Develops some theories of objective probability to give what Pollock calls “nomic probability,” which is a certain version of chance given by physical laws. Argues for various epistemic connections between this nomic probability and rational degrees of belief in a form that is strikingly different from more traditional accounts.

                                                                                                                                                              Find this resource:

                                                                                                                                                              Subjective and Objective Bayesianism

                                                                                                                                                              Among Bayesians, one of the central disagreements has been whether there are further constraints on degrees of belief beyond the probability axioms and update by conditionalization. Those who think there is a specific rationally mandated set of degrees of belief to have are known as “objective Bayesians,” while those who think all probability functions are permissible are known as “subjective Bayesians,” with those in the middle grouped with one or the other group depending on the context. Much of the discussion has turned on “Bertrand’s Paradox,” which is a series of problems posed in Bertrand 1907 that show that the most obvious principles for constraining rational degree of belief lead to contradictions. Jaynes 1973 argues that when a problem is posed correctly, only one of the constraints is actually applicable, so the problems can be avoided. Seidenfeld 1979 points out that even the notion of well-posedness in Jaynes 1973 fails to select a unique probability function in certain circumstances. De Finetti 1964 argues forcefully for the subjectivist position, and Jeffrey 1992 gives a motivation for another version of it. Carnap 1962 tries to develop a more logical version of probability, based on the syntax of language, but the author was never able to find a unique probability function that could be justified. A sophisticated modern defense of the approach in Carnap 1962 is given in Maher 2010.

                                                                                                                                                              • Bertrand, J. Calcul des probabilités. Paris: Gauthier-Villars, 1907.

                                                                                                                                                                Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                First published in 1888. The famous paradoxes are given on pp. 4–7. The rest of the book is a standard treatise on the theory of probability as developed to that date. No English translation of the whole volume has been identified, but the mathematical French is not very difficult for the pages cited.

                                                                                                                                                                Find this resource:

                                                                                                                                                                • Carnap, Rudolf. The Logical Foundations of Probability. Chicago: University of Chicago Press, 1962.

                                                                                                                                                                  Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                  Develops two concepts of probability. Argues that the concept connected to reasoning is a form of logic and gives several attempts at developing a mathematical theory of the “inductive probabilities” that arise as a result.

                                                                                                                                                                  Find this resource:

                                                                                                                                                                  • de Finetti, Bruno. “Foresight: Its Logical Laws, Its Subjective Sources.” In Studies in Subjective Probability. Edited by Henry Kyburg and H. E. Smokler, 93–158. Translated by Henry Kyburg. New York: Wiley, 1964.

                                                                                                                                                                    Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                    English translation of La prévision: Ses lois logique, ses sources subjectives (Paris: Institut Henri Poincaré), originally published in 1937. Defends a very subjective view of probability and argues that the only constraints are the axioms of probability.

                                                                                                                                                                    Find this resource:

                                                                                                                                                                    • Jaynes, Edwin T. “The Well-Posed Problem.” Foundations of Physics 3.4 (1973): 477–493.

                                                                                                                                                                      DOI: 10.1007/BF00709116Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                      Argues that the paradox in Bertrand 1907 can be solved by imposing an invariance requirement on the distribution. For each experimental situation, some salient invariance requirement allows one to find a unique probability distribution that fits, which can then be used as the prior. Available online for purchase or by subscription.

                                                                                                                                                                      Find this resource:

                                                                                                                                                                      • Jeffrey, Richard. “Radical Probabilism (Prospectus for a User’s Manual).” Philosophical Issues 2 (1992): 193–204.

                                                                                                                                                                        DOI: 10.2307/1522862Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                        Discusses Jeffrey’s “radical probabilism,” which is the view that there are no certainties and that all belief is essentially probabilistic. This rules out the possibility of any sort of certainty that could underlie an objective probability function. Available online by subscription.

                                                                                                                                                                        Find this resource:

                                                                                                                                                                        • Maher, Patrick. “Explication of Inductive Probability.” Journal of Philosophical Logic 39 (2010): 593–616.

                                                                                                                                                                          DOI: 10.1007/s10992-010-9144-4Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                          Shows that some problems for the inductive probabilities in Carnap 1962 were overestimated and gives a defense of the principles underlying them. Argues that this notion of inductive probability is what degree of belief should track.

                                                                                                                                                                          Find this resource:

                                                                                                                                                                          • Seidenfeld, Teddy. “Why I Am Not an Objective Bayesian.” Theory and Decision 11 (1979): 413–440.

                                                                                                                                                                            DOI: 10.1007/BF00139451Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                            Attacks objective Bayesians on a variety of issues. Shows that sometimes we have “too many” invariance requirements and that the invariance principles (including maximum entropy, which depends on invariance) are incompatible with conditionalization. Available online for purchase or by subscription.

                                                                                                                                                                            Find this resource:

                                                                                                                                                                            Bayesian Philosophy of Science

                                                                                                                                                                            Although many contemporary philosophers of science are focused on questions about the history of science and the actual practices of particular sciences, many Bayesians continue earlier traditions about trying to understand what is common to all sciences considered abstractly. The central idea is that notions of degree of belief can be used to understand the important notions of evidence, justification, and confirmation that are of central importance in science. Hosiasson-Lindenbaum 1940 gives one of the earliest Bayesian approaches to these topics, showing how to get around some paradoxes that had recently been proposed for other formal approaches. Howson and Urbach 2006 summarizes much of the Bayesian program. Earman 1992 shows how the Bayesian approach deals with not only many issues in the philosophy of science, but also raises some serious worries about it. Extensive additional discussion of these and related topics can be found in the article on Confirmation.

                                                                                                                                                                            • Earman, John. Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory. Cambridge, MA: MIT Press, 1992.

                                                                                                                                                                              Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                              Gives a history of Bayes’ theorem and the growth of its importance in the philosophy of science. Shows how Bayesianism answers some traditional questions in philosophy of science. Raises several important problems for Bayesianism. Ends with a discussion of some alternatives.

                                                                                                                                                                              Find this resource:

                                                                                                                                                                              • Hosiasson-Lindenbaum, Janina. “On Confirmation.” Journal of Symbolic Logic 5 (1940): 133–148.

                                                                                                                                                                                DOI: 10.2307/2268173Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                Develops much of the formal technique of Bayesian confirmation theory. Shows that it can help explain many traditional ideas about induction without running into some of the paradoxes. In particular, shows that by distinguishing the absolute lack of confirmation from exceedingly small amounts of confirmation, some paradoxical intuitions can be revised. Available online by subscription.

                                                                                                                                                                                Find this resource:

                                                                                                                                                                                • Howson, Colin, and Peter Urbach. Scientific Reasoning: The Bayesian Approach. 3d ed. Chicago: Open Court, 2006.

                                                                                                                                                                                  Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                  A manifesto for the Bayesian approach to the philosophy of science. Chapter 4 of the third edition gives approaches to many classic problems of induction from the Bayesian standpoint.

                                                                                                                                                                                  Find this resource:

                                                                                                                                                                                  Old Evidence, Logical Omniscience, New Theories

                                                                                                                                                                                  Three related, important problems for Bayesianism are the Problem of Old Evidence, the Problem of Logical Omniscience, and the Problem of New Theories. The Problem of Old Evidence was raised in Glymour 1981. The author points out that in the Bayesian picture, anything that is already certain cannot provide confirmation in the future. However, many historical successes in science, such as Einstein’s explanation of the perihelion shift of Mercury, take precisely this form of using old evidence as new confirmation. Garber 1983 argues that it is not the old evidence that provides the confirmation, but rather learning of a logical relation between a theory and the old evidence. This requires a form of Bayesianism in which logical facts are not required to already be old information. Attempts to develop such theories have been given in Hacking 1967 and, in more detail, in Gaifman 2004, though all such attempts are still incomplete. Eells 1985 classifies problems of old evidence to show that while some aspects of the problem are related to logical omniscience, others are due to the fact that the theory to be confirmed was proposed after the evidence was discovered. Chihara 1987 suggests that Bayesianism will have trouble allowing for the proposal of new theories, but Maher 1995 gives an approach to the problem. Some of these issues, especially ones connected to the Problem of Old Evidence, are discussed further in the article on Evidence.

                                                                                                                                                                                  • Chihara, Charles. “Some Problems for Bayesian Confirmation Theory.” British Journal for the Philosophy of Science 38.4 (1987): 551–560.

                                                                                                                                                                                    DOI: 10.1093/bjps/38.4.551Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                    Describes an example situation in which some evidence seems irrelevant to a conclusion for a long time, but the proposal of a new theory suddenly changes the confirmatory power of old evidence, which must require a change of credences by means other than conditionalization. Available online for purchase or by subscription.

                                                                                                                                                                                    Find this resource:

                                                                                                                                                                                    • Eells, Ellery. “Problems of Old Evidence.” Pacific Philosophical Quarterly 66 (1985): 283–302.

                                                                                                                                                                                      Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                      Classifies instances of the Problem of Old Evidence in Glymour 1981 into several types. Shows that the solution in Garber 1983 addresses some types but not others. Raises worries about how to measure evidential relations among one’s current beliefs, which don’t seem to be addressed by confirmation theory.

                                                                                                                                                                                      Find this resource:

                                                                                                                                                                                      • Gaifman, Haim. “Reasoning with Limited Resources and Assigning Probabilities to Arithmetical Statements.” Synthese 140.1–2 (2004): 97–119.

                                                                                                                                                                                        DOI: 10.1023/B:SYNT.0000029944.99888.a7Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                        Gives a formal theory of probability over the language of arithmetic and shows how to weaken the axioms so that they do not imply omniscience over all logical relations. In particular, only entailments that are provable in some fragment of the language will be required to be known. Available online for purchase or by subscription.

                                                                                                                                                                                        Find this resource:

                                                                                                                                                                                        • Garber, Dan. “Old Evidence and Logical Omniscience in Bayesian Confirmation Theory.” In Testing Scientific Theories. Edited by John Earman, 99–132. Minneapolis: University of Minnesota Press, 1983.

                                                                                                                                                                                          Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                          Argues that the confirmation in the Problem of Old Evidence is actually given by a new logical fact that is learned, not the old empirical fact. Gives a first attempt at developing a theory of probability that can accommodate logical learning, though the restrictions imposed are somewhat artificial.

                                                                                                                                                                                          Find this resource:

                                                                                                                                                                                          • Glymour, Clark. “Why I Am Not a Bayesian.” In Theory and Evidence. Edited by Clark Glymour, 63–93. Chicago: University of Chicago Press, 1981.

                                                                                                                                                                                            Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                            Develops many critiques of Bayesianism, including the new Problem of Old Evidence. Argues that philosophers accept Bayesianism only because of its successes, and that these successes all depend on ad hoc manipulation of the probability function rather than being derivable from first principles.

                                                                                                                                                                                            Find this resource:

                                                                                                                                                                                            • Hacking, Ian. “Slightly More Realistic Personal Probability.” Philosophy of Science 34.4 (1967): 311–325.

                                                                                                                                                                                              DOI: 10.1086/288169Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                              Points out that traditional Bayesianism requires agents to be aware of all logical consequences. Proposes a modification of the theory that weakens this requirement but doesn’t give a full mathematical investigation of what the resulting theory would look like. Available online by subscription.

                                                                                                                                                                                              Find this resource:

                                                                                                                                                                                              • Maher, Patrick. “Probabilities for New Theories.” Philosophical Studies 77 (1995): 103–115.

                                                                                                                                                                                                DOI: 10.1007/BF00996314Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                Develops a way for Bayesians to assign probabilities to theories that haven’t yet been proposed. This approach is metalinguistic and assigns probabilities to the truth of theories under various descriptions rather than to the theories. Available online for purchase or by subscription.

                                                                                                                                                                                                Find this resource:

                                                                                                                                                                                                Bayesianism outside Philosophy

                                                                                                                                                                                                Bayesianism has been influential in many fields outside of philosophy as well. This article cannot do justice to the vast array of ways in which the program has been used, but it can provide the beginnings of an outline. This influence is most direct in the field of statistics, which consists of the use of probabilities for purposes of inference. Bayesians have argued for a drastic reformulation of the foundations of statistics, which requires different inference techniques than the traditional ones. Gelman, et al. 2004 provides a textbook introduction to the methods of Bayesian statistics. Royall 1997 provides critiques of traditional statistical methodology and shows that Bayesian methodology has a better foundation. Gelman and Shalizi 2012 argues that, although Bayesian methods are often better than the traditional ones, the Bayesian foundations still leave much to be desired. In legal thinking, Bayesianism has become important in understanding how juries should reason about evidence. Finkelstein and Fairley 1970 argues that juries should be educated in the proper use of Bayes’ theorem so that they don’t over- or under-estimate the relevance of psychologically compelling pieces of evidence. In response, Tribe 1971 argues that this would introduce new problems of misuse of legal evidence. In psychology, much controversy has arisen over the extent to which the mind can be understood as operating on Bayesian principles. Tversky and Kahneman 1974 demonstrates a series of ways in which actual reasoners fail to reason in accord with these rules. But Griffiths, et al. 2008 argues that, nevertheless, Bayesianism is very useful for modeling cognition at a variety of levels.

                                                                                                                                                                                                • Finkelstein, Michael, and William Fairley. “A Bayesian Approach to Identification Evidence.” Harvard Law Review 83 (1970): 489–517.

                                                                                                                                                                                                  DOI: 10.2307/1339656Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                  Argues that juries often ignore evidence about background probabilities in favor of evidence that appears to be relevant to a specific case. Shows that, in some such cases, explicit tutoring in how to use Bayesian techniques can give lead to more appropriate outcomes. Available online by subscription.

                                                                                                                                                                                                  Find this resource:

                                                                                                                                                                                                  • Gelman, Andrew, and Cosma Shalizi. “Philosophy and the Practice of Bayesian Statistics.” In Oxford Handbook of the Philosophy of Social Science. Edited by Harold Kincaid, 259–273. Oxford: Oxford University Press, 2012.

                                                                                                                                                                                                    Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                    Argues that although Bayesian statistical techniques can be useful, the Bayesian foundation for them is faulty. Statistical inference should be understood along the lines of Popperian hypothetico-deductivism rather than induction from priors to posteriors.

                                                                                                                                                                                                    Find this resource:

                                                                                                                                                                                                    • Gelman, Andrew, John Carlin, Hal Stern, and Donald Rubin. Bayesian Data Analysis. Boca Raton, FL: Chapman and Hall, 2004.

                                                                                                                                                                                                      Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                      Provides a background for statisticians in how to work with Bayesian methods.

                                                                                                                                                                                                      Find this resource:

                                                                                                                                                                                                      • Griffiths, Thomas, Charles Kemp, and Joshua Tenenbaum. “Bayesian Models of Cognition.” In Cambridge Handbook of Computational Psychology. Edited by Ron Sun, 59–100. Cambridge, UK: Cambridge University Press, 2008.

                                                                                                                                                                                                        DOI: 10.1017/CBO9780511816772Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                        Surveys Bayesian learning techniques. Shows how many of them can be used to understand various psychological processes. Summarizes much of the literature on these applications.

                                                                                                                                                                                                        Find this resource:

                                                                                                                                                                                                        • Royall, Richard. Statistical Evidence: A Likelihood Paradigm. Boca Raton, FL: Chapman and Hall, 1997.

                                                                                                                                                                                                          Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                          Argues that the traditional foundations for statistics make many conceptual errors, and that Bayesian foundations are more coherent. However, because of the subjectivity of priors, argues for the use of likelihood methods, which can use the Bayesian foundations.

                                                                                                                                                                                                          Find this resource:

                                                                                                                                                                                                          • Tribe, Laurence. “Trial by Mathematics: Precision and Ritual in the Legal Process.” Harvard Law Review 84 (1971): 1329–1393.

                                                                                                                                                                                                            DOI: 10.2307/1339610Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                            Shows that although Bayesian techniques can give more accurate results in specific cases, they can lead to a focus on the wrong kind of evidence. In particular, some evidence that is required for Bayesian techniques is ruled inadmissible on moral and legal grounds, while other evidence that is legally relevant cannot be given with the kind of mathematical precision needed for Bayesianism. Available online by subscription.

                                                                                                                                                                                                            Find this resource:

                                                                                                                                                                                                            • Tversky, Amos, and Daniel Kahneman. “Judgment under Uncertainty: Heuristics and Biases.” Science 185.4157 (1974): 1124–1131.

                                                                                                                                                                                                              DOI: 10.1126/science.185.4157.1124Save Citation »Export Citation »E-mail Citation »

                                                                                                                                                                                                              Summarizes the results of several experiments showing ways in which actual humans deviate from ideal Bayesian rationality in reasoning. Argues that these deviations are the result of cognitive heuristics that can work in many cases, but that can lead us astray in others. Available online for purchase or by subscription.

                                                                                                                                                                                                              Find this resource:

                                                                                                                                                                                                              back to top

                                                                                                                                                                                                              Article

                                                                                                                                                                                                              Up

                                                                                                                                                                                                              Down