Bayesian Epistemology (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeBayesian EpistemologyFirst published Thu Jul 12, 2001; substantive revision Wed Mar 26, 2008‘Bayesian epistemology’ became an epistemological movementin the 20th century, though its two main features can betraced back to the eponymous Reverend Thomas Bayes (c. 1701-61). Thosetwo features are: (1) the introduction of a formal apparatusfor inductive logic; (2) the introduction of a pragmaticself-defeat test (as illustrated by Dutch Book Arguments) forepistemic rationality as a way of extending the justificationof the laws of deductive logic to include a justification for the lawsof inductive logic. The formal apparatus itself has two main elements:the use of the laws of probability as coherence constraints onrational degrees of belief (or degrees of confidence) and theintroduction of a rule of probabilistic inference, a rule or principleof conditionalization.Bayesian epistemology did not emerge as a philosophical programuntil the first formal axiomatizations of probability theory in thefirst half of the 20th century. One importantapplication of Bayesian epistemology has been to the analysis ofscientific practice in Bayesian Confirmation Theory. Inaddition, a major branch of statistics, Bayesianstatistics, is based on Bayesian principles. In psychology,an important branch of learning theory, Bayesian learningtheory, is also based on Bayesian principles. Finally, theidea of analyzing rational degrees of belief in terms of rationalbetting behavior led to the 20th century development of anew kind of decision theory, Bayesian decision theory, which isnow the dominant theoretical model for the both the descriptive andnormative analysis of decisions. The combination of its preciseformal apparatus and its novel pragmatic self-defeat test forjustification makes Bayesian epistemology one of the most importantdevelopments in epistemology in the 20th century, and one ofthe most promising avenues for further progress in epistemology in the21st century. 1. Deductive and Probabilistic Coherence and Deductive and Probabilistic Rules of Inference2. A Simple Principle of Conditionalization3. Dutch Book Arguments4. Bayes' Theorem and Bayesian Confirmation Theory Bayes' Theorem and a Corollary Bayesian Confirmation Theory 5. Bayesian Social Epistemology6. Potential Problems 6.1 Objections to the Probability Laws as Standards of Synchronic Coherence 6.2 Objections to The Simple Principle of Conditionalization as a Rule of Inference and Other Objections to Bayesian Confirmation Theory 7. Other Principles of Bayesian EpistemologyBibliographyOther Internet ResourcesRelated Entries1. Deductive and Probabilistic Coherence and Deductive and Probabilistic Rules of InferenceThere are two ways that the laws of deductive logic have been thoughtto provide rational constraints on belief: (1) Synchronically, thelaws of deductive logic can be used to define the notion of deductiveconsistency and inconsistency. Deductive inconsistency so defineddetermines one kind of incoherence in belief, which I refer to asdeductive incoherence. (2) Diachronically, the laws ofdeductive logic can constrain admissible changes in belief byproviding the deductive rules of inference. For example,modus ponens is a deductive rule of inference that requiresthat one infer Q from premises P and P → Q.Bayesians propose additional standards of synchronic coherence —standards of probabilistic coherence — and additional rulesof inference — probabilistic rules of inference — in bothcases, to apply not to beliefs, but degrees of belief (degrees ofconfidence). For Bayesians, the most important standards ofprobabilistic coherence are the laws of probability. For more on thelaws of probability, see the following supplementary article: Supplement on Probability LawsFor Bayesians, the most important probabilistic rule of inference isgiven by a principle of conditionalization. 2. A Simple Principle of ConditionalizationIf unconditional probabilities (e.g. P(S)) are takenas primitive, the conditional probability of S on Tcan be defined as follows:Conditional Probability: P(S/T) = P(S&T)/P(T).By itself, the definition of conditional probability is of littleepistemological significance. It acquires epistemologicalsignificance only in conjunction with a further epistemologicalassumption:Simple Principle of Conditionalization: If one begins with initial or prior probabilitiesPi, and one acquires new evidence whichcan be represented as becoming certain of an evidentiary statementE (assumed to state the totality of one's new evidenceand to have initial probability greater than zero), then rationalityrequires that one systematically transform one's initialprobabilities to generate final or posterior probabilitiesPf by conditionalizing on E —that is: Where S is any statement,Pf(S) = Pi(S/E).[1] In epistemological terms, this Simple Principle of Conditionalizationrequires that the effects of evidence on rational degrees be analyzedin two stages: The first is non-inferential. It is the change in theprobability of the evidence statement E fromPi(E), assumed to be greaterthan zero and less than one, toPf(E) = 1. The second is aprobabilistic inference of conditionalizing on E from initialprobabilities (e.g., Pi(S)) tofinal probabilities (e.g., Pf(S)= Pi(S/E)).Problems with the Simple Principle (to be discussed below) have ledmany Bayesians to qualify the Simple Principle by limiting itsscope. In addition, some Bayesians follow Jeffrey in generalizing theSimple Principle to apply to cases in which one's new evidence is lessthan certain (also discussed below). What unifies Bayesianepistemology is a conviction that conditionalizing (perhaps of ageneralized sort) is rationally required in some important contexts— that is, that some sort of conditionalization principle is animportant principle governing rational changes in degrees ofbelief.3. Dutch Book ArgumentsMany arguments have been given for regarding the probability laws ascoherence conditions on degrees of belief and for taking someprinciple of conditionalization to be a rule of probabilisticinference. The most distinctively Bayesian are those referred to asDutch Book Arguments. Dutch Book Arguments represent thepossibility of a new kind of justification for epistemologicalprinciples.A Dutch Book Argument relies on some descriptive or normativeassumptions to connect degrees of belief with willingness to wager —for example, a person with degree of belief p in sentenceS is assumed to be willing to pay up to and including$p for a unit wager on S (i.e., a wager that pays $1if S is true) and is willing to sell such a wager for anyprice equal to or greater than $p (one is assumed to be equallywilling to buy or sell such a wager when the price is exactly $p).[2] A Dutch Book is a combination ofwagers which, on the basis of deductive logic alone, can be shown toentail a sure loss. A synchronic Dutch Book is a Dutch Bookcombination of wagers that one would accept all at the same time. Adiachronic Dutch Book is a Dutch Book combination of wagersthat one will be motivated to enter into at different times. Ramsey and de Finetti first employed synchronic Dutch Book Argumentsin support of the probability laws as standards of synchroniccoherence for degrees of belief. The first diachronic Dutch BookArgument in support of a principle of conditionalization was reportedby Teller, who credited David Lewis. The Lewis/Teller argument dependson a further descriptive or normative assumption about conditionalprobabilities due to de Finetti: An agent with conditional probabilityP(S/T) = p is assumed to bewilling to pay any price up to and including $p for a unitwager on S conditional on T. (A unit wager onS conditional on T is one that is called off, withthe purchase price returned to the purchaser, if T is nottrue. If T is true, the wager is not called off and the wagerpays $1 if S is also true.) On this interpretation ofconditional probabilities, Lewis, as reported by Teller, was able toshow how to construct a diachronic Dutch Book against anyone who, onlearning only that T, would predictably change his/her degreeof belief in S to Pf(S)> Pi(S/T); and howto construct a diachronic Dutch Book against anyone who, on learningonly that T, would predictably change his/her degree ofbelief in S to Pf(S)< Pi(S/T). Forillustrations of the strategy of the Ramsey/de Finetti and theLewis/Teller arguments, see the following supplementary article: Supplement on Dutch Book ArgumentsThere has been much discussion of exactly what it is that Dutch BookArguments are supposed to show. On the literal-mindedinterpretation, their significance is that they show that thosewhose degrees of belief violate the probability laws or those whoseprobabilistic inferences predictably violate a principle ofconditionalization are liable to enter into wagers on which they aresure to lose. There is very little to be said for the literal-mindedinterpretation, because there is no basis for claiming thatrationality requires that one be willing to wager in accordance withthe behavioral assumptions described above. An agent could simplyrefuse to accept Dutch Book combinations of wagers.One of the main motivations for Jeffrey's new approach to thefoundations of decision theory in Logic of Decision was hisdissatisfaction with the identification of subjective probability withbetting ratios. For example, no matter what one's degree of belief inthe proposition that all human life will be destroyed within the nextten years, it would be not be rational to offer to buy a bet on itstruth. Williamson extends de Finetti's Dutch Book Argument for afinite additivity constraint on rational degrees of belief to producean argument for a countable additivity constraint on degrees ofbelief, but the argument is better interpreted as a reductio of theliteral-minded interpretation of Dutch Book Arguments than as anargument for the rationality of a countable additivity constraint.The rational response to offers to bet on the proposition that alllife will be destroyed within the next ten years or to bet on a singlepossible outcome in a countably infinite set of equiprobable possibleoutcomes is simply not to.A more plausible interpretation of Dutch Book Arguments is that theyare to be understood hypothetically, as symptomatic of what has beentermed pragmatic self-defeat. On this interpretation, DutchBook Arguments are a kind of heuristic for determining whenone's degrees of belief have the potential to bepragmatically self-defeating. The problem is not that onewho violates the Bayesian constraints is likely to enter into acombination of wagers that constitute a Dutch Book, but that, on anyreasonable way of translating one's degrees of belief intoaction, there is a potential for one's degrees of belief tomotivate one to act in ways that make things worse than they mighthave been, when, as a matter of logic alone, it can be determinedthat alternative actions would have made things better (on one'sown evaluations of better and worse). Another way of understanding the problem of susceptibility to a DutchBook is due to Ramsey: Someone who is susceptible to a Dutch Bookevaluates identical bets differently based on how they are described.Putting it this way makes susceptibility to Dutch Books soundirrational. But this standard of rationality would make itirrational not to recognize all the logical consequences of what onebelieves. This is the assumption of logical omniscience(discussed below). If successful, Dutch Book Arguments would reduce the justificationof the principles of Bayesian epistemology to two elements: (1) anaccount of the appropriate relationship between degrees of belief andchoice; and (2) the laws of deductive logic. Because it would seemthat the truth about the appropriate relationship between the degreesof belief and choice is independent of epistemology, Dutch BookArguments hold out the potential of justifying the principles ofBayesian epistemology in a way that requires no other epistemologicalresources than the laws of deductive logic. For this reason, it makessense to think of Dutch Book Arguments as indirect, pragmaticarguments for according the principles of Bayesian epistemology muchthe same epistemological status as the laws of deductive logic.Dutch Book Arguments are a truly distinctive contribution made byBayesians to the methodology of epistemology. It should also be mentioned that some Bayesians have defended theirprinciples more directly, with non-pragmatic arguments. In additionto reporting Lewis's Dutch Book Argument, Teller offers anon-pragmatic defense of Conditionalization. There have been manyproposed non-pragmatic defenses of the probability laws (e.g., vanFraassen; Shimony). The most compelling is due to Joyce. All suchdefenses, whether pragmatic or non-pragmatic, produce a puzzle forBayesian epistemology: The principles of Bayesian epistemology aretypically proposed as principles of inductive reasoning. Butif the principles of Bayesian epistemology depend ultimately for theirjustification solely on the laws of deductive logic, what reason isthere to think that they have any inductive content? That isto say, what reason is there to believe that they do anything morethan extend the laws of deductive logic from beliefs to degrees ofbelief? It should be mentioned, however, that even if Bayesianepistemology only extended the laws of deductive logic to degrees ofbelief, that alone would represent an extremely important advance inepistemology. 4. Bayes' Theorem and Bayesian Confirmation TheoryThis section reviews some of the most important results in theBayesian analysis of scientific practice — Bayesian ConfirmationTheory. It is assumed that all statements to be evaluated haveprior probability greater than zero and less than one.Bayes' Theorem and a CorollaryBayes' Theorem is a straightforward consequence of the probabilityaxioms and the definition of conditional probability:Bayes' Theorem: P(S/T) = P(T/S) ×P(S)/P(T) [whereP(T) is assumed to be greater than zero] The epistemological significance of Bayes' Theorem is that itprovides a straightforward corollary to the Simple Principle ofConditionalization. Where the final probability of a hypothesisH is generated by conditionalizing on evidence E,Bayes' Theorem provides a formula for the final probability ofH in terms of the prior or initial likelihood ofH on E(Pi(E/H)) and the prioror initial probabilities of H and E: Corollary of the Simple Principle ofConditionalization:Pf(H) =Pi(H/E) =Pi(E/H) ×Pi(H)/Pi(E). Due to the influence of Bayesianism, likelihood is now atechnical term of art in confirmation theory. As used in thistechnical sense, likelihoods can be very useful. Often, when theconditional probability of H on E is in doubt, thelikelihood of H on E can be computed from thetheoretical assumptions of H.Bayesian Confirmation TheoryA. Confirmation and disconfirmation. In BayesianConfirmation Theory, it is said that evidence confirms (or wouldconfirm) hypothesis H (to at least some degree) just in casethe prior probability of H conditional on E isgreater than the prior unconditional probability of H:Pi(H/E) >Pi(H). E disconfirms(or would disconfirm) H if the prior probability ofH conditional on E is less than the priorunconditional probability of H.This is a qualitative conception of confirmation. There is no generalagreement in the literature on a quantitative measure of degree ofconfirmation or degree of evidential support. Earman (chap. 5) andFitelson both provide a good overview of the various proposals. Itmight be thought that the degree to which evidence E supports (orwould support) hypothesis H could be defined asPi(H/E) −Pi(H). One potential problemwith this proposal is that it has the consequence that no evidence canprovide much evidential support to a hypothesis that is antecedentlyvery probable, because as the probability of H approachesone, the difference goes to zero. Eells and Fitelson have argued thatthis apparently counterintuitive consequence can be avoided bydistinguishing the historical question of how much a piece of evidenceE actually contributed to the confirmation of H(which, of course, would have to be small if H were antecedentlyhighly probable) from the question of the degree of evidential supportE provides for H, the answer to which, they propose,is relative to the background information. So even if H isvery probable at the time that evidence E is acquired, we canask how much evidential support E would provide forH if we had no other evidence supporting H. Eellsand Fitelson have also provided a useful framework for evaluating thevarious proposals in the literature, a framework within which most ofthem are found to be wanting. B. Confirmation and disconfirmation by entailment.Whenever a hypothesis H logically entails evidenceE, E confirms H. This follows from the factthat to determine the truth of E is to rule out a possibilityassumed to have non-zero prior probability that is incompatible withH — the possibility that ~E. A corollary isthat, where H entails E, ~E woulddisconfirm H, by reducing its probability to zero. The mostinfluential model of explanation in science is thehypothetico-deductive model (e.g., Hempel). Thus, one of the mostimportant sources of support for Bayesian Confirmation Theory is thatit can explain the role of hypothetico-deductive explanation inconfirmation. C. Confirmation of logical equivalents. If twohypotheses H1 and H2 are logically equivalent, then evidenceE will confirm both equally. This follows from the fact thatlogically equivalent statements always are assigned the sameprobability.D. The confirmatory effect of surprising or diverseevidence. From the corollary above, it follows that whetherE confirms (or disconfirms) H depends on whetherE is more probable (or less probable) conditional onH than it is unconditionally — that is, on whether: (b1) P(E/H)/P(E) > 1.An intuitive way of understanding (b1) is to say that it states thatE would be more expected (or less surprising) if it wereknown that H were true. So if E is surprising, butwould not be surprising if we knew H were true, thenE will significantly confirm H. Thus, Bayesiansexplain the tendency of surprising evidence to confirm hypotheses onwhich the evidence would be expected.Similarly, because it is reasonable to think that evidenceE1 makes other evidence of the same kind much moreprobable, after E1 has been determined to be true,other evidence of the same kind E2 will generallynot confirm hypothesis H as much as other diverse evidenceE3, even if H is equally likely on bothE2 and E3. The explanation isthat where E1 makes E2 muchmore probable than E3(Pi(E2/E1)>>Pi(E3/E1),there is less potential for the discovery that E2is true to raise the probability of H than there is for thediscovery that E3 is true to do so. E. Relative confirmation and likelihood ratios. Oftenit is important to be able to compare the effect of evidenceE on two competing hypotheses,Hj and Hk, withouthaving also to consider its effect on other hypotheses that may not beso easy to formulate or to compare withHj and Hk.From the first corollary above, the ratio of the final probabilitiesof Hj and Hkwould be given by:Ratio Formula: Pf(Hj)/Pf(Hk)=[Pi(E/Hj) × Pi(Hj)]/[Pi(E/Hk) ×Pi(Hk)]If the odds of Hj relative toHk are defined as ratio of their probabilities, then fromthe Ratio Formula it follows that, in a case in which change indegrees of belief results from conditionalizing on E, thefinal odds(Pf(Hj)/Pf(Hk))result from multiplying the initial odds(Pi(Hj)/Pi(Hk))by the likelihood ratio(Pi(E/Hj)/Pi(E/Hk)). Thus,in pairwise comparisons of the odds of hypotheses, the likelihoodratio is the crucial determinant of the effect of the evidence on theodds.F. Subjective and Objective Bayesianism. Are thereconstraints on prior probabilities other than the probability laws?Consider a situation in which you are to draw a ball from an urnfilled with red and black balls. Suppose you have no otherinformation about the urn. What is the prior probability (beforedrawing a ball) that, given that a ball is drawn from the urn, thatthe drawn ball will be black? The question divides Bayesians into twocamps:(a) Subjective Bayesians emphasize the relative lack ofrational constraints on prior probabilities. In the urn example, theywould allow that any prior probability between 0 and 1 might berational (though some Subjective Bayesians (e.g., Jeffrey) would ruleout the two extreme values, 0 and 1). The most extreme SubjectiveBayesians (e.g., de Finetti) hold that the only rational constraint onprior probabilities is probabilistic coherence. Others (e.g.,Jeffrey) classify themselves as subjectivists even though they allowfor some relatively small number of additional rational constraints onprior probabilities. Since subjectivists can disagree aboutparticular constraints, what unites them is that their constraintsrule out very little. For Subjective Bayesians, our actual priorprobability assignments are largely the result of non-rationalfactors—for example, our own unconstrained, free choice orevolution or socialization.(b) Objective Bayesians (e.g., Jaynes and Rosenkrantz)emphasize the extent to which prior probabilities are rationallyconstrained. In the above example, they would hold that rationalityrequires assigning a prior probability of 1/2 to drawing a black ballfrom the urn. They would argue that any other probability would failthe following test: Since you have no information at all about whichballs are red and which balls are black, you must choose priorprobabilities that are invariant with a change in label (“red” or“black”). But the only prior probability assignment that is invariantin this way is the assignment of prior probability of 1/2 to each ofthe two possibilities (i.e., that the ball drawn is black or that itis red).In the limit, an Objective Bayesian would hold that rationalconstraints uniquely determine prior probabilities in everycircumstance. This would make the prior probabilities logicalprobabilities determinable purely a priori. None ofthose who identify themselves as Objective Bayesians holds thisextreme form of the view. Nor do they all agree on precisely what therational constraints on degrees of belief are. For example,Williamson does not accept Conditionalization in any form as arational constraint on degrees of belief. What unites all of theObjective Bayesians is their conviction that in many circumstances,symmetry considerations uniquely determine the relevant priorprobabilities and that even when they don't uniquely determine therelevant prior probabilities, they often so constrain the range ofrationally admissible prior probabilities, as to assure convergence onthe relevant posterior probabilities. Jaynes identifies four generalprinciples that constrain prior probabilities, group invariance,maximium entropy, marginalization, and coding theory, but he does notconsider the list exhaustive. He expects additional principles to beadded in the future. However, no Objective Bayesian claims that thereare principles that uniquely determine rational prior probabilities inall cases.By introducing symmetry constraints on prior probabilities, theObjective Bayesians inherit the difficulties of the classicalPrinciple of Indifference, so-named by Keynes, but usually attributedto Laplace. The simple example of the urn illustrates how invarianceconsiderations can be used to give content to the Principle ofIndifference. There the objectivist is able to uniquely determine theprior probabilities from the requirement that the rational priorprobabilities should be invariant under switching the labels used toclassify the balls in the urn.However, it is generally agreed by both objectivists and subjectiviststhat ignorance alone cannot be the basis for assigning priorprobabilities. The reason is that in any particular case there mustbe some information to pick out which parameters or whichtransformations are the ones among which one is to be indifferent.Without such information, indifference considerations lead to paradox.Objective Bayesians have been quite creative in finding ways toresolve many of the paradoxes (e.g., Jeffreys' solution to Bertrand'sPardox, Jaynes's solution to Buffon's Needle Paradox, or Mikkelson'ssolution to van Mises' Paradox). But there are always more paradoxes.Charles, Höcker, Lacker, Le Diberder, and T'Jampens provide anactual example from physics where maximum entropy yields conflictingresults depending on parameterization and where a frequentist approachseems to be superior to any Objective Bayesian approach that employsany form of Conditionalization.G. The typical differential effect of positive evidence andnegative evidence. Hempel first pointed out that we typicallyexpect the hypothesis that all ravens are black to be confirmed tosome degree by the observation of a black raven, but not by theobservation of a non-black, non-raven. Let H be thehypothesis that all ravens are black. Let E1describe the observation of a non-black, non-raven. LetE2 describe the observation of a blackraven. Bayesian Confirmation Theory actually holds that bothE1 and E2 may provide someconfirmation for H. Recall that E1supports H just in casePi(E1/H)/Pi(E1)> 1. It is plausible to think that this ratio is ever so slightlygreater than one. On the other hand, E2 would seem toprovide much greater confirmation to H, because, in this example, itwould be expected thatPi(E2/H)/Pi(E2)>>Pi(E1/H)/Pi(E1). These are only a sample of the results that have provided supportfor Bayesian Confirmation Theory as a theory of rational inference forscience. For further examples, see Howson and Urbach. Itshould also be mentioned that an important branch of statistics,Bayesian statistics is based on the principles of Bayesianepistemology. 5. Bayesian Social EpistemologyOne of the important developments in Bayesian epistemology has beenthe exploration of the social dimension to inquiry. The obviousexample is scientific inquiry, because it is the community ofscientists, rather than any individual scientist, who determine whatis or is not accepted in the discipline. In addition, scientiststypically work in research groups and even those who work alone relyon the reports of other scientists to be able to design and carry outtheir own work. Other important examples of the social dimension toknowledge include the use of juries to make factual determinations inthe legal system and the decentralization of knowledge over theInternet.There are two ways that Bayesian epistemology can be applied to socialinquiry:(1) Bayesian epistemology of testimony (understood generally, toinclude not only personal testimony but all media sources ofinformation). Goldman has developed a Bayesian epistemology oftestimony and applied it to social entities such as science and thelegal system. In any such approach, a crucial issue is how toevaluate the reliability of the reports one receives. Goldman'sapproach is to focus on institutional design to motivate theproduction of reliable reports. Bovens and Hartmann instead try tomodel how, when there are reports from multiple sources, a Bayesianagent can use probabilistic reasoning to judge the reliability of thereports, and thus, how much credence to place in them. The idea thatin evaluating the probability of a report we are implicitly evaluatingthe reliability of the reporter is developed by Barnes as a potentialexplanation of the prediction/accommodation asymmetry, discussed inthe next section.(2) Aggregate Bayesianism. If scientific knowledge or jurydeliberations produce a group product, it is natural to considerwhether the group's knowledge can be represented in aggregate form.In Bayesian terms, the question is whether the individuals'probabililty assignments can be usefully aggregated into a singleprobability assignment that reflects the group's knowledge. AlthoughSeidenfeld, Kadane, and Schervish have shown that there is generallyno way to define an aggregate Bayesian expected utility maximizer torepresent the Pareto preferences of a group of two or more individualBayesian expected utility maximizers, there is no impossibility resultprecluding the aggregation of individual probabililty assignments intoa group probability assignment. However, there is no generally agreedupon rule for doing so. If a group of Bayesian individuals all hadbegun from the same initial probabilities, then simply sharing theirevidence would lead them all to the same final probabilities. It mayseem unfortunate that unanimity in science and other social endeavorscannot be achieved so easily, but Kitcher has argued that this is amistake, because cognitive diversity plays an important role inscientific progress.The fruitfulness of Bayesian social epistemology may ultimatelydepend on whether or not the idealizations of Bayesian theory are toounrealistic. For example, if one of the important effects of jurydeliberations is that they tend to provide a way for the group tocorrect for the irrationality of individual members, then no model ofjurors as ideal Bayesians is likely to be able to explain that featureof the jury system.6. Potential ProblemsThis section reviews some of the most important potential problemsfor Bayesian Confirmation Theory and for Bayesian epistemologygenerally. No attempt is made to evaluate their seriousness here,though there is no generally agreed upon Bayesian solution to any ofthem.6.1 Objections to the Probability Laws as Standards of Synchronic CoherenceA. The assumption of logical omniscience. Theassumption that degrees of belief satisfy the probability laws impliesomniscience about deductive logic, because the probability lawsrequire that all deductive logical truths have probability one, alldeductive inconsistencies have probability zero, and the probabilityof any conjunction of sentences be no greater than any of itsdeductive consequences. This seems to be an unrealistic standard forhuman beings. Hacking and Garber have made proposals to relax theassumption of logical omniscience. Because relaxing that assumptionwould block the derivation of almost all the important results inBayesian epistemology, most Bayesians maintain the assumption oflogical omniscience and treat it as an ideal to which human beings canonly more or less approximate.B. The special epistemological status of the laws ofclassical logic. Even if the assumption of logicalomniscience is not too much of an idealization to provide a usefulmodel for human reasoning, it has another potentially troublingconsequence. It commits Bayesian epistemology to some sort of apriori/a posteriori distinction, because there could be no Bayesianaccount of how empirical evidence might make it rational to adopt atheory with a non-classical logic. In this respect, Bayesianepistemology carries over the presumption from traditionalepistemology that the laws of logic are immune to revision on thebasis of empirical evidence.It is open to the Bayesian to try to downplay the significance ofthis consequence, by articulating an a priori/a posteriori distinctionthat aims to be pragmatic rather than metaphysical (e.g., Carnap'sanalytic/synthetic distinction). However, any such account mustaddress Quine's well-known holistic challenge to theanalytic-synthetic distinction.6.2 Objections to The Simple Principle of Conditionalization as a Rule of Inference and Other Objections to Bayesian Confirmation TheoryA. The problem of uncertain evidence. The SimplePrinciple of Conditionalization requires that the acquisition ofevidence be representable as changing one's degree of belief in astatement E to one — that is, to certainty. But manyphilosophers would object to assigning probability of one to anycontingent statement, even an evidential statement, because, forexample, it is well-known that scientists sometimes give up previouslyaccepted evidence. Jeffrey has proposed a generalization of thePrinciple of Conditionalization that yields that principle as aspecial case. Jeffrey's idea is that what is crucial aboutobservation is not that it yields certainty, but that it generates anon-inferential change in the probability of an evidential statementE and its negation ~E (assumed to be the locus ofall the non-inferential changes in probability) from initialprobabilities between zero and one toPf(E) andPf(~E) = [1 −Pf(E)]. Then onJeffrey's account, after the observation, the rational degree ofbelief to place in an hypothesis H would be given by thefollowing principle:Principle of Jeffrey Conditionalization: Pf(H) = Pi(H/E) × Pf(E) + Pi(H/~E) ×Pf(~E) [where E and H are both assumed to have priorprobabilities between zero and one]Counting in favor of Jeffrey's Principle is its theoreticalelegance. Counting against it is the practical problem that itrequires that one be able to completely specify the directnon-inferential effects of an observation, something it is doubtfulthat anyone has ever done. Skyrms has given it a Dutch Book defense.B. The problem of old evidence. On a Bayesianaccount, the effect of evidence E in confirming (ordisconfirming) a hypothesis is solely a function of the increase inprobability that accrues to E when it is first determined tobe true. This raises the following puzzle for Bayesian ConfirmationTheory discussed extensively by Glymour: Suppose that E is anevidentiary statement that has been known for some time — thatis, that it is old evidence; and suppose that H is ascientific theory that has been under consideration for some time. Oneday it is discovered that H implies E. In scientificpractice, the discovery that H implied E wouldtypically be taken to provide some degree of confirmatory support forH. But Bayesian Confirmation Theory seems unable to explainhow a previously known evidentiary statement E could provideany new support for H. For conditionalization to come into play, theremust be a change in the probability of the evidence statementE. Where E is old evidence, there is no change inits probability. Some Bayesians who have tried to solve this problem(e.g., Garber) have typically tried to weaken the logical omniscienceassumption to allow for the possibility of discovering logicalrelations (e.g., that H and suitable auxiliary assumptionsimply E). As mentioned above, relaxing the logicalomniscience assumption threatens to block the derivation of almost allof the important results in Bayesian epistemology. Other Bayesians(e.g., Lange) employ the Bayesian formalism as a tool in therational reconstruction of the evidentiary support for ascientific hypothesis, where it is irrelevant to the rationalreconstruction whether the evidence was discovered before or after thetheory was initially formulated. Joyce and Christensen agree thatdiscovering new logical relations between previously accepted evidenceand a theory cannot raise the probability of the theory. However,they suggest that usingPi(H/E) −Pi(H/-E) as a measureof support can at least explain how evidence that has probability onecould still support a theory. Eells and Fitelson have criticized thisproposal and argued that the problem is better addressed bydistinguishing two measures, the historical measure of the degree towhich a piece of evidence E actually confirmed an hypothesisH and the ahistorical measure of how much a piece of evidenceE would support an hypothesis H, on given backgroundinformation B. The second measure enables us to ask theahistorical question of how much E would support Hif we had no other evidence supporting H. C. The problem of rigid conditional probabilities.When one conditionalizes, one applies the initial conditionalprobabilities to determine final unconditional probabilities.Throughout, the conditional probabilities themselves do not change;they remain rigid. Examples of the Problem of Old Evidence are butone of a variety of cases in which it seems that it can be rationalto change one's initial conditional probabilities. Thus, manyBayesians reject the Simple Principle of Conditionalization in favorof a qualified principle, limited to situations in which one does notchange one's initial conditional probabilities. There is nogenerally accepted account of when it is rational to maintain rigidinitial conditional probabilities and when it is not.D. The problem of prediction vs. accommodation.Related to the problem of Old Evidence is the following potentialproblem: Consider two different scenarios. In the first, theoryH was developed in part to accommodate (i.e., toimply) some previously known evidence E. In the second, theoryH was developed at a time when E was not known. Itwas because E was derived as a prediction fromH that a test was performed and E was found to betrue. It seems that E's being true would provide a greater degree ofconfirmation for H if the truth of E had beenpredicted by H than if H had been developedto accommodate the truth of E. There is no generalagreement among Bayesians about how to resolve this problem. Some(e.g., Horwich) argue that Bayesianism implies that there is noimportant difference between prediction and accommodation, and try todefend that implication. Others (e.g., Maher) argue that there is away to understand Bayesianism so as to explain why there is animportant difference between prediction and accommodation. E. The problem of new theories. Suppose that there isone theory H1 that is generally regarded as highlyconfirmed by the available evidence E. It is possible thatsimply the introduction of an alternative theoryH2 can lead to an erosion ofH1's support. It is plausible to think thatCopernicus' introduction of the heliocentric hypothesis had thiseffect on the previously unchallenged Ptolemaic earth-centeredastronomy. This sort of change cannot be explained byconditionalization. It is for this reason that many Bayesians preferto focus on probability ratios of hypotheses (see the Ratio Formulaabove), rather than their absolute probability; but it is clear thatthe introduction of a new theory could also alter the probabilityratio of two hypotheses — for example, if it implied one of themas a special case.F. The problem of the priors. Are there constraintson prior probabilities other than the probability laws? This is theissue that divides the Subjective from the Objective Bayesians, asdiscussed above. Consider Goodman's “new riddle of induction”: In thepast all observed emeralds have been green. Do those observationsprovide any more support for the generalization that all emeralds aregreen than they do for the generalization that all emeralds are grue(green if observed before now; blue if observed later); or do theyprovide any more support for the prediction that the next emeraldobserved will be green than for the prediction that the next emeraldobserved will be grue (i.e., blue)? Almost everyone agrees that itwould be irrational to have prior probabilities that were indifferentbetween green and grue, and thus made predictions of greeness no moreprobable than predictions of grueness. But there is no generallyagreed upon explanation of this constraint.The problem of the priors identifies an important issue between theSubjective and Objective Bayesians. If the constraints on rationalinference are so weak as to permit any or almost any probabilisticallycoherent prior probabilities, then there would be nothing to makeinferences in the sciences any more rational than inferences inastrology or phrenology or in the conspiracy reasoning of a paranoidschizophrenic, because all of them can be reconstructed as inferencesfrom probabilistically coherent prior probabilities. Some SubjectiveBayesians believe that their position is not objectionably subjective,because of results (e.g., Doob or Gaifman and Snir) proving that evensubjects beginning with very different prior probabilities will tendto converge in their final probabilities, given a suitably long seriesof shared observations. These convergence results are not completelyreassuring, however, because they only apply to agents who alreadyhave significant agreement in their priors and they do not assureconvergence in any reasonable amount of time. Also, they typicallyonly guarantee convergence on the probability of predictions, not onthe probability of theoretical hypotheses. For example, Carnap favoredprior probabilities that would never raise above zero the probabilityof a generalization over a potentially infinite number of instances(e.g., that all crows are black), no matter how many observations ofpositive instances (e.g., black crows) one might make without findingany negative instances (i.e., non-black crows). In addition, theconvergence results depend on the assumption that the only changes inprobabilities that occur are those that are the non-inferentialresults of observation on evidential statements and those that resultfrom conditionalization on such evidential statements. But almost allsubjectivists allow that it can sometimes be rational to change one'sprior probability assignments.Because there is no generally agreed upon solution to the Problem ofthe Priors, it is an open question whether Bayesian ConfirmationTheory has inductive content, or whether it merely translates theframework for rational belief provided by deductive logic into acorresponding framework for rational degrees of belief.7. Other Principles of Bayesian EpistemologyOther principles of Bayesian epistemology have been proposed, butnone has garnered anywhere near a majority of support amongBayesians. The most important proposals are merely mentionedhere. It is beyond the scope of this entry to discuss them in anydetail.A. Other principles of synchronic coherence. Are theprobability laws the only standards of synchronic coherence fordegrees of belief? Van Fraassen has proposed an additional principle(Reflection or Special Reflection), which he now regards as a specialcase of an even more general principle (General Reflection).[3] B. Other probabilistic rules of inference. There seemto be at least two different concepts of probability: the probabilitythat is involved in degrees of belief (epistemic or subjectiveprobability) and the probability that is involved in random events,such as the tossing of a coin (chance). De Finetti thought this was amistake and that there was only one kind of probability, subjectiveprobability. For Bayesians who believe in both kinds of probability,an important question is: What is (or should be) the relation betweenthem? The answer can be found in the various proposals for principlesof direct inference in the literature. Typically, principles ofdirect inference are proposed as principles for inferring subjectiveor epistemic probabilities from beliefs about objective chance (e.g.,Pollock). Lewis reverses the direction of inference, and proposes toinfer beliefs about objective chance from subjective or epistemicprobabilities, via his (Reformulated) Principal Principle.[4] Strevens argues that it is Lewis's Principal Principle that givesBayesianism its inductive content.C. Principles of rational acceptance. What is therelation between beliefs and degrees of belief? Jeffrey proposes togive up the notion of belief (at least for empirical statements) andmake do with only degrees of belief. Other authors (e.g., Levi,Maher, Kaplan) propose principles of rational acceptance as part ofaccounts of when it is rational to accept a statement as true, notmerely to regard it as probable.BibliographyBarnes, Eric Christian, “Predictivism for Pluralists”, BritishJournal for the Philosophy of Science 56 (2005): 421-450.Bayes, Thomas, “An Essay Towards Solving a Problem in the Doctrineof Chances”, Philosophical Transactions of the Royal Society ofLondon (1764) 53: 37-418, reprinted in E.S. Pearson and M.G.Kendall, eds., Studies in the History of Statistics and probability(London: Charles Griffin, 1970).Bovens, Luc, and Stephan Hartmann, Bayesian Epistemology(Oxford: Clarendon Press; 2003).Carnap, Rudolf, Logical Foundations of Probability(Chicago: University of Chicago Press; 1950).Carnap, Rudolf, The Continuum of Inductive Methods(Chicago: University of Chicago Press; 1952).Carnap, “Meaning Postulates”, in Meaning and Necessity(Chicago: Phoenix Books; 1956): 222-229.Charles, J., Hocker, A., Lacker, H., Le Diberder, F.R., T'Jampens,S., “Bayesian Statistics at Work: the Troublesome Extraction of theCKM Phase alpha”, URL:http://aps.arxiv.org/abs/hep-ph/0607246(7/22/2006).Christensen, David, Putting Logic in its Place: FormalConstraints on Rational Belief (Oxford: Clarendon Press;2004).Christensen, David, “Measuring Confirmation,” Journal ofPhilosophy 96 (1999): 437-461.de Finetti, Bruno, “La Prevision: ses lois logiques, se sourcessubjectives” (Annales de l'Institut Henri Poincare 7 (1937): 1-68.Translated into English and reprinted in Kyburg and Smokler,Studies in Subjective Probability (Huntington, NY: Krieger;1980).Doob, J.L., “What is a Martingale?”, American MathematicalMonthly 78 (1971): 451-462.Earman, John, Bayes or Bust? A Critical Examination of BayesianConfirmation Theory (Cambridge, MA: MIT Press; 1992).Eells, Ellery, and Branden Fitelson. 2000. “Measuring Confirmationand Evidence”, Journal of Philosophy 97: 663-672.Eells, Ellery, and Branden Fitelson. 2002. “Symmetries andAsymmetries in Evidential Support”, Philosophical Studies107: 129-142.Fitelson, Branden. 1999. “The Plurality of Bayesian Measures of Confirmationand the Problem of Measure Sensitivity,” Philosophy of Science 66 (ProceedingsSupplement): S362-378.Fitelson, Branden. 2003. “Review of James Joyce, TheFoundations of Causal Decision Theory”, in Mind 112:545-551.Gaifman, H., and Snir, M., “Probabilities over Rich Languages”,Journal of Symbolic Logic 47 (1982): 495-548.Garber, Daniel, “Old Evidence and Logical Omniscience in BayesianConfirmation Theory”, in J. Earman, ed., Testing ScientificTheories, Midwest Studies in the Philosophy of Science,Vol. X (Minneapolis: University of Minnesota Press; 1983):99-131.Goldman, Alvin I., Knowledge in a Social World (Oxford:Clarendon Press; 1999).Goodman, Nelson, Fact, Fiction, and Forecast (Cambridge:Harvard University Press; 1983).Glymour, Clark, Theory and Evidence (Princeton: PrincetonUniversity Press; 1980).Hacking, Ian, “Slightly More Realistic Personal Probability”,Philosophy of Science 34 (1967): 311-325.Hempel, Carl G., Aspects of Scientific Explanation (NewYork: Free Press; 1965).Horwich, Paul, Probability and Evidence (Cambridge:Cambridge University Press; 1982).Howson, Colin, and Peter Urbach, Scientific Reasoning: TheBayesian Approach, 2nd ed. (Chicago: Open Court; 1993).Jaynes, E.T., “Prior Probabilities”, Institute of Electricaland Electronic Engineers Transactions on Systems Science andCybernetics, SSC-4 (1968): 227-241.Jaynes, E.T. (ed. by G. Larry Bretthorst), Probability Theory:The Logic of Science (Cambridge: Cambridge University Press;2003).Jeffrey, Richard, The Logic of Decision, 2nd ed. (Chicago:University of Chicago Press; 1983).Jeffrey, Richard, Probability and the Art of Judgment(Cambridge: Cambridge University Press; 1992).Jeffreys, Harold, Theory of Probability, 3d ed. (Oxford:Clarendon Press; ([1948] 1961).Joyce, James M., “A Nonpragmatic Vindication of Probabilism”,Philosophy of Science 65 (1998): 575-603.Joyce, James M., The Foundations of Causal DecisionTheory (Cambridge: Cambridge University Press; 1999).Kaplan, Mark, Decision Theory as Philosophy (Cambridge:Cambridge University Press; 1996).Keynes, John Maynard, A Treatise on Probability (London:Macmillan; 1921).Kitcher, Philip, “The Division of Cognitive Labor”, Journal ofPhilosophy 87 (1990): 5-22.Lange, Marc, “Calibration and the Epistemological Role of BayesianConditionalization”, Journal of Philosophy 96 (1999):294-324.Laplace, P. S. Marquis de, Théorie Analytique desProbabilitis, 3d ed. (Paris: Gauthier-Villars; ([1820]1886).Levi, Isaac, The Enterprise of Knowledge (Cambridge,Mass.: MIT Press; 1980)Levi, Isaac, The Fixation Of Belief And Its Undoing(Cambridge: Cambridge University Press; 1991).Lewis, David, “A Subjectivist's Guide to Objective Chance”, inRichard C. Jeffrey, ed., Studies in Inductive Logic andProbability, vol. 2 (Berkeley: University of California Press;1980): 263-293.Maher, Patrick, “Prediction, Accommodation, and the Logic ofDiscovery”, PSA, vol. 1 (1988): 273-285.Maher, Patrick, Betting on Theories (Cambridge: CambridgeUniversity Press; 1993).Mikkelson, Jeffrey M., “Dissolving the Wine/Water Paradox”,British Journal for the Philosophy of Science 55 (2004):137-145.Pollock, John L., Nomic Probability and the Foundations ofInduction (Oxford: Oxford University Press; 1990).Popper, Karl, The Logic of Scientific Discovery,3rd ed. (London: Hutchinson; 1968).Quine, W.V., “Carnap on Logical Truth”, in The Ways ofParadox (New York: Random House; 1966): 100-125.Ramsey, Frank P., “Truth and Probability,” in Richard B.Braithwaite (ed.), Foundations of Mathematics and Other LogicalEssay (London: Routledge and Kegan Paul; Check on 1931publication date), pp. 156-198.Réyni, A., “On a New Axiomatic Theory of Probability”,Acta Mathematica Academiae Scientiarium Hungaricae 6 (1955):285-385.Rosenkrantz, R.D., Foundations and Applications of InductiveProbability (Atascadero, CA: Ridgeview Publishing; 1981).Savage, Leonard, The Foundations of Statistics, 2nd ed.(New York: Dover; 1972).Seidenfeld, Teddy, Joseph B. Kadane, and Mark J. Schervish, “Onthe Shared Preferences of Two Bayesian Decision Makers”, Journalof Philosophy 86 (1989): 225-244.Shimony, Abner, “An Adamite Derivation of the Calculus ofProbability”, in J.H. Fetzer, ed., Probability and Causalty(Dordrecht: Reidel; 1988).Skyrms, Brian, Pragmatics and Empiricism (New Haven: YaleUniversity Press; 1984).Skyrms, Brian, TheDynamics of Rational Deliberation(Cambridge, Mass.: Harvard University Press; 1990).Sober, Elliott, “Bayesianism—Its Scope and Limits”, in RichardSwinburne, ed., Bayes's Theorem (Oxford: Oxford UniversityPress; 2002): 21-38.Strevens, Michael, “Bayesian Confirmation Theory: Inductive Logic,or Mere Inductive Framework?”, Synthese 141 (2004):365-379.Teller, Paul, “Conditionalization, Observation, and Change ofPreference”, in W. Harper and C.A. Hooker, eds., Foundations ofProbability Theory, Statistical Inference, and Statistical Theories ofScience (Dordrecht: D. Reidel; 1976).Van Fraassen, Bas C., “Calibration: A Frequency Justification forPersonal Probability”, in R.S. Cohen and L. Laudan, eds., Physics,Philosophy, and Psychoanalysis: Essays in Honor of Adolf Grunbaum(Dordrecht: Reidel; 1983).Van Fraassen, Bas C., “Belief and the Will”, Journal ofPhilosophy 81 (1984): 235-256.Van Fraassen, Bas C., “Belief and the Problem of Ulysses and theSirens”, Philosophical Studies 77 (1995): 7-37.Williamson, Jon, “Countable Additivity and SubjectiveProbability”, Brit. J. Phil. Sci. 50 (1999): 401-416.Williamson, Jon, “Motivating Objective Bayesianism: From EmpiricalConstraints to Objective Probabilities,” in W. E. Harper andG. R. Wheeler, eds., Probability and Inference: Essays in Honourof Henry E. Kyburg, Jr. (Amsterdam: Elsevier; 2007).Zynda, Lyle, “Old Evidence and New Theories”, PhilosophicalStudies 77 (1995): 67-95.Other Internet Resources[Please contact the author with suggestions]Related Entries Bayes' Theorem | logic: inductive | probability, interpretations ofAcknowledgments In the preparation of this article, I have benefited from commentsfrom Marc Lange, Stephen Glaister, Laurence BonJour, and JamesJoyce. Copyright © 2008 byWilliam Talbott<wtalbott@u.washington.edu> |
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