Bayesian Epistemology

First published Mon Jun 13, 2022

We can think of belief as an all-or-nothing affair. For example, Ibelieve that I am alive, and I don’t believe that I am ahistorian of the Mongol Empire. However, often we want to makedistinctions betweenhow strongly we believe or disbelievesomething. I strongly believe that I am alive, am fairly confidentthat I will stay alive until my next conference presentation, lessconfident that the presentation will go well, and strongly disbelievethat its topic will concern the rise and fall of the Mongol Empire.The idea that beliefs can come in different strengths is a centralidea behind Bayesian epistemology. Such strengths are calleddegrees of belief, orcredences. Bayesianepistemologists study norms governing degrees of beliefs, includinghow one’s degrees of belief ought to change in response to avarying body of evidence. Bayesian epistemology has a long history.Some of its core ideas can be identified in Bayes’ (1763)seminal paper in statistics (Earman 1992: ch. 1), with applicationsthat are now very influential in many areas of philosophy and ofscience.

The present entry focuses on the more traditional, general issuesabout Bayesian epistemology, and, along the way, interested readerswill be referred to entries that discuss the more specific topics. Atutorial on Bayesian epistemology will be provided in the firstsection for beginners and those who want a quick overview.

1. A Tutorial on Bayesian Epistemology

This section provides an introductory tutorial on Bayesianepistemology, with references to subsequent sections or relatedentries for details.

1.1 A Case Study

For a glimpse of what Bayesian epistemology is, let’s see whatBayesians have to say about this episode in scientific inquiry:

Example (Eddington’s Observation).Einstein’s theory of General Relativity entails that light canbe deflected by a massive body such as the Sun. This physical effect,predicted by Einstein in a 1911 paper, was observed during a solareclipse on May 29, 1919, from locations chosen from Eddington’stwo expeditions. This result surprised the physics community and wasdeemed a significant confirmation of Einstein’s theory.

The above case makes a general point:

The Principle of Hypothetico-DeductiveConfirmation. Suppose that a scientist is testing a hypothesisH. She deduces from it an empirical consequenceE, anddoes an experiment, being not sure whetherE is true. It turnsout that she obtainsE as new evidence as a result of theexperiment. Then she ought to become more confident inH.Moreover, the more surprising the evidenceE is, the higher thecredence inH ought to be raised.

This intuition about how credences ought to change can be vindicatedin Bayesian epistemology by appeal to two norms. But before turning tothem, we need a setting. Divide the space of possibilities into four,according to whether hypothesisH is true or false and whetherevidenceE is true or false. SinceH logically impliesE, there are only three distinct possibilities on the table,which are depicted as the three dots infigure 1.

a diagram: link to extended description below

Figure 1: A Space of ThreePossibilities. [Anextended description of figure 1.]

Those possibilities aremutually exclusive in the sense thatno two of them can hold together; and they arejointlyexhaustive in the sense that at least one of them must hold. Aperson can be more or less confident that a given possibility holds.Suppose that it makes sense to say of a person that she is, say, 80%confident that a certain possibility holds. In this case, say thatthis person’s degree of belief, or credence, in that possibilityis equal to 0.8. A credence might be any other real number. (How tomake sense of real-valued credences is a major topic for Bayesians, tobe discussed in§1.6 and§1.7 below.)

Now I can sketch the two core norms in Bayesian epistemology.According to the first norm, calledProbabilism, one’scredences in the three possibilities infigure 1 ought to fit together so nicely that they are non-negative and sum to1. Such a distribution of credences can be represented by a bar chart,as depicted on the left offigure 2.

Figure 2: Conditionalization onEvidence. [Anextended description of figure 2.]

Now, suppose that a person with this credence distribution receivesE as new evidence. It seems that as a result, there should besome change in credences. But how should they change? According to thesecond norm, called thePrinciple of Conditionalization, thepossibility incompatible withE (i.e., the rightmostpossibility) should have its credence dropped down to 0, and tosatisfy Probabilism, the remaining credences should be scaledup—rescaled to sum to 1. So this person’s credence inhypothesisH has to rise in a way such as that depicted infigure 2.

Moreover, suppose that new evidenceE is very surprising. Itmeans that the person starts out being highly confident in the falsityofE, as depicted on the left offigure 3.

Figure 3: Conditionalization onSurprising Evidence. [Anextended description of figure 3.]

Then conditionalization onE requires a total credence collapsefollowed by a dramatic scaling-up of the other credences. Inparticular, the credence inH is raised significantly, unlessit is zero to begin with. This vindicates the intuition reported inthe case of Eddington’s Observation.

1.2 Two Core Norms

The two Bayesian norms sketched above can be stated a bit moregenerally as follows. (A formal statement will be provided after thistutorial, insection 2.) Suppose that there are some possibilities under consideration, whichare mutually exclusive and jointly exhaustive. A proposition underconsideration is one that is true or false in each of thosepossibilities, so it can be identified with the set of thepossibilities in which it is true. When those possibilities are finitein number, and when you have credences in all of them, Probabilismtakes a simple form, saying that your credences ought to beprobabilistic in this sense:

(Non-Negativity) The credences assigned to thepossibilities under consideration are non-negative real numbers.
(Sum-to-One) The credences assigned to thepossibilities under consideration sum to 1.
(Additivity) The credence assigned to aproposition under consideration is equal to the sum of the credencesassigned to the possibilities in that proposition.

While this norm issynchronic in that it constrains yourcredences at each time, the next norm isdiachronic. Supposethat you just received a piece of evidenceE, which is true inat least some possibilities under consideration. Suppose further thatE exhausts all the evidence you just received. Then thePrinciple of Conditionalization says that your credences ought tochange as if you followed the procedure below (although it is possibleto design other procedures to the same effect):

(Zeroing) For each possibility incompatiblewith evidenceE, drop its credence down to zero.
(Rescaling) For the possibilities compatiblewith evidenceE, rescale their credences by a common factor tomake them sum to 1.
(Resetting) Now that there is a new credencedistribution over the individual possibilities, reset the credences inpropositions according to the Additivity rule in Probabilism.

The second step, rescaling, deserves attention. It is designed toensure compliance with Probabilism, but it also has an independent,intuitive appeal. Consider any two possibilities in which new evidenceE is true. Thus the new evidence alone cannot distinguish thosetwo possibilities and, hence, it seems to favor the two equally. So itseems that, if a person starts out being twice as confident in one ofthose two possibilities as in the other, she should remain so afterthe credence change in light ofE, as required by the rescalingstep. The essence of conditionalization is preservation of certainratios of credences, which is a feature inherited by generalizationsof conditionalization (seesection 5 for details).

So there you have it: Probabilism and the Principle ofConditionalization, which are held by most Bayesians to be the twocore norms in Bayesian epistemology.

1.3 Applications

Bayesian epistemology features an ambition: to develop a simplenormative framework that consists of little or nothing more than thetwo core Bayesian norms, with the goal of explaining or justifying awide range of intuitively good epistemic practices and perhaps also ofguiding our inquiries, all done with a focus on credence change. Thatsounds quite ambitious, given the narrow focus on credence change. Butmany Bayesians maintain that credence change is a unifying theme thatunderlies many different aspects of our epistemic endeavors. Let memention some examples below.

First of all, it seems that a hypothesisH isconfirmed by new evidenceE exactly when one’scredence inH ought to increase in response to the acquisitionofE. Extending that idea, it also seems thathow muchH is confirmed correlates with how much its credence ought tobe raised. With those ideas in mind, Bayesians have developed severalaccounts of confirmation; seesection 3 of the entry on confirmation. Through the concept of confirmation, some Bayesians have alsodeveloped accounts of closely related concepts. For example, beingsupported by evidence seems to be the same as or similar tobeing confirmed by evidence, which is ultimately explained byBayesians in terms of credence change. So there are some Bayesianaccounts of evidential support; seesection 3 of the entry on Bayes’ theorem andsections 2.3–2.5 of the entry on imprecise probabilities. Here is another example:how well a theoryexplainsa body of evidence seems to be closely related to how well the theoryis confirmed by the evidence, which is ultimately explained byBayesians in terms of credence change. So there are some Bayesianaccounts of explanatory power; seesection 2 of the entry on abduction.

The focus on credence change also sheds light on another aspect of ourepistemic practices: inductive inference. An inductive inference isoften understood as a process that results in the formation of anall-or-nothing attitude: believing or accepting the truth of ahypothesisH on the basis of one’s evidenceE.That does not appear to fit the Bayesian picture well. But toBayesians, what really matters is how new evidenceE ought tochange one’s credence inH—whether one’scredence ought to beraised orlowered, and byhow much. To be sure, there is the issue of whether theresulting credence would be high enough to warrant the formation ofthe attitude of believing or accepting. But to many Bayesians, thatissue seems only secondary, or better forgone as argued by Jeffrey(1970). If so, the fundamental issue about inductive inference isultimately how credences ought to change in light of new evidence. SoBayesians have had much to say about various kinds of inductiveinferences and related classic problems in philosophy of science. Seethe following footnote for a long list of relevant survey articles (orresearch papers, in cases where survey articles are not yet available).^[1]

For monographs on applications in epistemology and philosophy ofscience, see Earman (1992), Bovens & Hartmann (2004), Howson &Urbach (2006), and Sprenger & Hartmann (2019). In fact, there arealso applications to natural language semantics and pragmatics: forindicative conditionals, see the survey by Briggs (2019: sec. 6 and 7)and sections 3 and 4.2 of the entry onindicative conditionals; for epistemic modals, see Yalcin (2012).

The applications mentioned above rely on the assumption of some orother norms for credences. Although the correct norms are held by mostBayesians to include at least Probabilism and the Principle ofConditionalization, it is debated whether there are more and, if so,what they are. It is to this issue that I now turn.

1.4 Bayesians Divided: What Does Coherence Require?

Probabilism is often regarded as acoherence norm, which sayshow one’s opinions ought to fit together on pain of incoherence.So, if Probabilism matters, the reason seems to be that coherencematters. This raises a question that divides Bayesians:What doesthe coherence of credences require? A typical Bayesian thinksthat coherence requires at least that one’s credences followProbabilism. But there are actually different versions of Probabilismand Bayesians disagree about which one is correct. Bayesians alsodisagree about whether the coherence of credences requires more thanProbabilism and, if so, to what extent. For example, does coherencerequire that one’s credence in acontingent propositionlie strictly between 0 and 1? Another issue is what coherence requiresof conditional credences, i.e., the credences that one has on thesupposition of the truth of one or another proposition. Those andother related questions have far-reaching impacts on applications ofBayesian epistemology. For more on the issue of what coherencerequires, seesection 3.

1.5 Bayesians Divided: The Problem of the Priors

There is another issue that divides Bayesians. The package ofProbabilism and the Principle of Conditionalization seems to explainwell why one’s credence in General Relativity ought to rise inEddington’s Observation Case. But that particular Bayesianexplanation relies on a crucial feature of the case: the evidenceE isentailed by the hypothesisH in question.But such an entailment is missing in many interesting cases, such asthis one:

Example (Enumerative Induction). After a dayof field research, we observed one hundred black ravens without acounterexample. So the newly acquired evidence isE = “wehave observed one hundred ravens and they all were black”. Weare interested in this hypothesisH = “the next raven tobe observed will be black”.

Now, should the credence in the hypothesis be increased or lowered,according to the two core Bayesian norms? Well, it depends. Note thatin the present caseH entails neitherE nor itsnegation, so the possibilities inH can be categorized into twogroups: those compatible withE, and those incompatible withE. As a result of conditionalization, the possibilitiesincompatible withE will have their credences be dropped downto zero; those compatible, scaled up. If the scaling up outweighs thedropping down for the possibilities insideH, the credence inH will rise and thus behave inductively; otherwise, it willstay constant or even go down and thus behave counter-inductively. Soit all depends on the specific details of theprior, which isshorthand for the assignment of credences that one has before oneacquires the new evidence in question. To sum up: Probabilism and thePrinciple of Conditionalization, alone, are too weak to entitle us tosay whether one’s credence ought to change inductively orcounter-inductively in the above example.

This point just made generalizes to most applications of Bayesianepistemology. For example, some coherent priors lead to enumerativeinduction and some don’t (Carnap 1955), and some coherent priorslead to Ockham’s razor and some don’t (Forster 1995: sec.3). So, besides the coherence norms (such as Probabilism), are thereany other norms that govern one’s prior? This is known asthe problem of the priors.

This issue divides Bayesians. First of all, there is the party ofsubjective Bayesians, who hold that every prior is permittedunless it fails to be coherent. So, to those Bayesians, the correctnorms for priors are exhausted by Probabilism and the other coherencenorms if any. Second, there is the party ofobjectiveBayesians, who propose that the correct norms for priors includenot just the coherence norms but also a norm that codifies theepistemic virtue of freedom from bias. Those Bayesians think thatfreedom from bias requires at least that, roughly speaking,one’s credences be evenly distributed to certain possibilitiesunless there is a reason not to. This norm, known asthe Principleof Indifference, has long been a source of controversy. Last butnot the least, some Bayesians even propose to take seriously certainepistemic virtues that have been extensively studied in otherepistemological traditions, and argue that those virtues need to becodified into norms for priors. For more on those attempted solutionsto the problem of the priors, seesection 4 below. Also seesection 3.3 of the entry on interpretations of probability.

So far I have been mostly taking for granted the package ofProbabilism and the Principle of Conditionalization. But is there anygood reason to accept those two norms? This is the next topic.

1.6 An Attempted Foundation: Dutch Book Arguments

There have been a number of arguments advanced in support of the twocore Bayesian norms. Perhaps the most influential is of the kindcalledDutch Book arguments. Dutch Book arguments aremotivated by a simple, intuitive idea: Belief guides action. So, themore strongly you believe that it will rain tomorrow, the moreinclined you are, or ought to be, to bet on bad weather. This idea,which connects degrees of belief to betting dispositions, can becaptured at least partially by the following:

A Credence-Betting Bridge Principle (ToyVersion). If one’s credence in a propositionA isequal to a real numbera, then it is acceptable for one to buythe bet “Win $100 ifA is true” at the price$\$100 \cdot a$ (and at any lower price).

This bridge principle might be construed as part of a definition or asa necessary truth that captures the nature of credences, or understoodas a norm that jointly constrains credences and betting dispositions(Christensen 1996; Pettigrew 2020a: sec. 3.1). The hope is that,through this bridge principle or perhaps a refined one, bad credencesgenerate bad symptoms in betting dispositions. If so, a close look atbetting dispositions might help us sort out bad credences from goodones. This is the strategy that underlies Dutch Book arguments.

To illustrate, consider an agent who has a .75 credence in propositionA and a .30 credence in its negation $\neg A$ (which violatesProbabilism). Assuming the bridge principle stated above, the agent iswilling to bet as follows:

Buy “win $100 ifA is true” at $\$75$.
Buy “win $100 if $\neg A$ is true” at $\$30$.

So the agent is willing to accepteach of those two offers.But it is actually very bad to acceptboth at the same time,for that leads to a sure loss (of $5):

	A is true	A is false
buy “win $100 ifA istrue” at $75	$-\$75 + \$100$	$-\$75$
buy “win $100 if $\neg A$ istrue” at $30	$-\$30$	$-\$30 + \$100$
net payoff	$-\$5$	$-\$5$

So this agent’s betting dispositions make her susceptible to aset of bets that are individually acceptable but jointly inflict asure loss. Such a set of bets is called aDutch Book. Theabove agent is susceptible to a Dutch Book, which sounds bad for theagent. So what has gone wrong? The problem seems to be this: Beliefguides action, and in this case, bad beliefs result in bad actions:garbage in, garbage out. Therefore, the agent should not have had thecombination of credence .75 in $A$ and .30 in $\neg A$ to beginwith—or so a Dutch Book argument would conclude.

The above line of thought can be generalized and turned into atemplate for Dutch Book arguments:

A Template for Dutch Book Arguments

Premise 1. You should follow such and such a credence-bettingbridge principle (or, due to the nature of credences, you do sonecessarily).
Premise 2. If you do, and if your credences violate constraintC, then provably you are susceptible to a Dutch Book.
Premise 3. But you should not be so susceptible.
Conclusion. So your credences should satisfy constraintC.

There is a Dutch Book argument for Probabilism (Ramsey 1926, deFinetti 1937). The idea can be extended to develop an argument for thePrinciple of Conditionalization (Lewis 1999, Teller 1973). Dutch Bookarguments have also been developed for other norms for credences, butthey require modifying the concept of a Dutch Book in one way oranother. Seesection 3 for references.

An immediate worry about Dutch Book arguments is that a highercredence might not be correlated with a stronger disposition to bet.Consider a person who loathes very much the anxiety caused by placinga bet. So, though she is very confident in a proposition, she mightstill refuse to buy a bet on its truth even at a low price—andrightly so. This seems to be a counterexample to premise 1 in theabove. For more on Dutch Book arguments, including objections to themas well as refinements of them, see the survey by Hájek (2009)and the entry onDutch Book arguments.

There is a notable worry that applies even if we have a Dutch Bookargument that is logically valid and only has true premises. A DutchBook argument seems to give only apractical reason foraccepting anepistemic norm: “Don’t have such andsuch combinations of credences, for otherwise there would be somethingbad pragmatically”. Such a reason seems unsatisfactory for thosewho wish to explain the correctness of the Bayesian norms with areason that is distinctively epistemic or at least non-pragmatic. SomeBayesians still think that Dutch Book arguments are good, and addressthe present worry by trying to give a non-pragmatic reformulation ofDutch Book arguments (Christensen 1996; Christensen 2004: sec. 5.3).Some other Bayesians abandon Dutch Book arguments and pursuealternative foundations of Bayesian epistemology, to which I turnnow.

1.7 Alternative Foundations

A second proposed type of foundation for Bayesian epistemology isbased on the idea ofaccurate estimation. This idea has twoparts: estimation, and its accuracy. On this approach, one’scredence in a propositionA is one’sestimate ofthe truth value ofA, whereA’s truth value isidentified with 1 if it is true and 0 if it is false (Jeffrey 1986).The closer one’s credence inA is to the truth value ofA, themore accurate one’s estimate is. Then aBayesian may argue that one’s credences ought to beprobabilistic, for otherwise the overall accuracy of one’scredence assignment would bedominated—namely, itwould, come what may, be lower than the overall accuracy of anothercredence assignment that one could have adopted. To some Bayesians,this gives a distinctively epistemic reason or explanation whyone’s credences ought to be probabilistic. The result is theso-calledaccuracy-dominance argument for Probabilism (Joyce1998). This approach has also been extended to argue for the Principleof Conditionalization (Briggs & Pettigrew 2020). For more on thisapproach, see the entry onepistemic utility arguments for probabilism as well as Pettigrew (2016).

There is a third proposed type of foundation for Bayesianepistemology. It appeals to a kind of doxastic state calledcomparative probability, which concerns a person’staking one proposition to bemore probable than, orasprobable as, orless probable than another proposition.On this approach, we postulate some bridge principles that connectone’s credences to one’s comparative probabilities. Hereis an example of such a bridge principle: for any propositionsX andY, ifX is equivalent to the disjunction oftwo incompatible propositions, each of which one takes to bemore probable thanY, then one’s credence inXshould be more thantwice of that inY. With suchbridge principles, a Bayesian may argue from norms for comparativeprobabilities to norms for credences, such as Probabilism. SeeFishburn (1986) for the historical development of this approach. SeeStefánsson (2017) for a recent defense and development. For ageneral survey of this approach, see Konek (2019). This approach hasbeen extended by Joyce (2003: sec. 4) to justify the Principle ofConditionalization.

The above are just some of the attempts to provide foundations forBayesian epistemology. For more, see the surveys by Weisberg (2011:sec. 4) and Easwaran (2011).

There is a distinctive class of worries for all the three proposedfoundations presented above, due to the fact that they rely on one oranother account of the nature of credences. This is where Bayesianepistemology meets philosophy of mind. Recall that they try tounderstand credences in relation to some other mental states: (i)betting dispositions, (ii) estimates of truth values, or (iii)comparative probabilities. But those accounts of credences areapparently vulnerable to counterexamples. (An example was mentionedabove: a person who dislikes the anxiety caused by betting seems to bea counterexample to the betting account of credences). For more onsuch worries, see Eriksson and Hájek (2007). For more onaccounts of credences, seesection 3.3 of the entry on interpretations of probability andsection 3.4 of the entry on imprecise probabilities.

There is a fourth,application-driven style of argument fornorms for credences that seems to be explicit or implicit in the mindsof many Bayesians. The idea is that a good argument for the two coreBayesian norms can be obtained by appealing to applications. The goalis to account for acomprehensive range of intuitively goodepistemic practices, all done with asimple set of generalnorms consisting of little or nothing more than the two core Bayesiannorms. If this Bayesian normative system is so good that, of the knowncompetitors, it strikes the best balance of those two virtues justmentioned—comprehensiveness and simplicity—thenthat is a good reason for accepting the two core Bayesiannorms. In fact, the method just described is applicable to any norm,for credences or for actions, in epistemology or in ethics. Somephilosophers argue that this method in its full generality, calledReflective Equilibrium, is the ultimate method for finding agood reason for or against norms (Goodman 1955; Rawls 1971). For moreon this method and its controversies, see the entry onreflective equilibrium.

The above are some ways to argue for Bayesian norms. The rest of thisintroductory tutorial is meant to sketch some general objections,leaving detailed discussions to subsequent sections.

1.8 Objections to Conditionalization

The Principle of Conditionalization requires one to react to newevidence by conditionalizing on it. So this principle, when construedliterally, appears to be silent on the case in which one receivesno new evidence. That is, it seems to be too weak to requirethat one shouldn’t arbitrarily change credences when there is nonew evidence. To remedy this, the Principle of Conditionalization isusually understood such that the case of no new evidence is identifiedwith the limiting case in which one acquires a logical truth astrivial new evidence, which rules out no possibilities. In that case,conditionalization on the trivial new evidence lowers no credences,and thus rescales credences only by a factor of 1—no credencechange at all—as desired. Once the Principle ofConditionalization is construed that way, it is no longer too weak,but then the worry is that it becomes too strong. Consider thefollowing case, which Earman (1992) adapts from Glymour (1980):

Example (Mercury). It is 1915. Einstein hasjust developed a new theory, General Relativity. He assesses the newtheory with respect to some old data that have been known for at leastfifty years: the anomalous rate of the advance of Mercury’sperihelion (which is the point on Mercury’s orbit that isclosest to the Sun). After some derivations and calculations, Einsteinsoon recognizes that his new theory entails the old data about theadvance of Mercury’s perihelion, while the Newtonian theory doesnot. Now, Einstein increases his credence in his new theory, andrightly so.

Note that, during his derivation and calculation, Einstein does notperform any experiment or collect any new astronomical data, so thebody of his evidence seems to remain unchanged, only consisting of theold data. Despite gaining no new evidence, Einstein changes (in fact,raises) his credence in the new theory, and rightly so—againstthe usual construal of the Principle of Conditionalization. Therefore,there is a dilemma for that principle: when construed literally, it istoo weak to prohibit arbitrary credence change; when construed in theusual way, it is too strong to accommodate Einstein’s credencechange in the Mercury Case. This problem isEarman’s problemof old evidence.

The problem of old evidence is sometimes presented in a differentway—in Glymour’s (1980) way—whose target of attackis not the Principle of Conditionalization but this:

Bayesian Confirmation Theory (A SimpleVersion). EvidenceE confirms hypothesisH for aperson at a time if and only if, at that time, her credence inH would be raised if she were to conditionalize onE(whether or not she actually does that).

IfE is an old piece of evidence that a person had receivedbefore, this person’s credence inE is currently 1. So,conditionalization onE at the present time would involvedropping no credence, followed by rescaling credences with a factor of1—so there is no credence change at all. Then, by the Bayesianaccount of confirmation stated above, old evidenceE must failto confirm new theoryH. But that result seems to be wrongbecause the old data about the advance of Mercury’s perihelionconfirmed Einstein’s new theory; this isGlymour’sproblem of old evidence, construed as a challenge to a Bayesianaccount of confirmation. But, if Earman (1992) is right, the MercuryCase challenges not just Bayesian confirmation theory, but actuallycuts deeper, all the way to one of the two core Bayesiannorms—namely, the Principle of Conditionalization—assuggested by Earman’s problem of old evidence. For attemptedsolutions to Earman’s old evidence problem (aboutconditionalization), seesection 5.1 below. For more on Glymour’s old evidence problem (aboutconfirmation), seesection 3.5 of the entry on confirmation.

The above is just the beginning of a series of problems for thePrinciple of Conditionalization, which will be discussed after thistutorial, insection 5. But here is a rough sketch: The problem of old evidence arises when anew theory is developed to accommodate some old evidence. When thefocus is shifted from old evidence to new theory, we shall discoveranother problem, no less thorny. Also note that the problem of oldevidence results from a kind of inflexibility in conditionalization:no credence change is permitted without new evidence. Additionalproblems have been directed at other kinds of inflexibility inconditionalization, such as the preservation of fully certaincredences. In response, some Bayesians defend the Principle ofConditionalization by trying to develop it into better versions, asyou will see insection 5.

1.9 Objections about Idealization

Another worry is that the two core Bayesian norms are not the kind ofnorms that we ought to follow, in that they are too demanding to beactually followed by ordinary human beings—after all,ought impliescan. More specifically, those Bayesiannorms are often thought to be too demanding for at least threereasons:

(Sharpness) Probabilism demands thatone’s credence in a proposition be extremely sharp, as sharp asan individual real number, precise to potentially infinitely manydigits.
(Perfect Fit) Probabilism demands thatone’s credences fit together nicely; for example, some credencesare required to sum to exactly 1, no more and no less—a perfectfit. The Principle of Conditionalization also demands a perfect fitamong three things: prior credences, posterior credences, and newevidence.
(Logical Omniscience) Probabilism is oftenthought to demand that one belogically omniscient, havingcredence 1 in every logical truth and credence 0 in every logicalfalsehood.

The last point, logical omniscience, might not be immediately clearfrom the preceding presentation, but it can be seen from thisobservation: A logical truth is true in all possibilities, so it hasto be assigned credence 1 by Sum-to-One and Additivity inProbabilism.

So the worry is that, although Bayesians have a simple normativeframework, they seem to enjoy the simplicity because they idealizeaway from the complications in humans’ epistemic endeavors andturn instead to normative standards that can be met only by highlyidealized agents. If so, there are pervasive counterexamples to thetwo core Bayesian norms: all human beings. Call thisthe problemof idealization. For different ways of presenting this problem,see Harman (1986: ch. 3), Foley (1992: sec. 4.4), Pollock (2006: ch.6), and Horgan (2017).

In reply, Bayesians have developed at least three strategies, whichmight complement each other. The first strategy is toremoveidealization gradually, one step at a time, and explain why this is agood way of doing epistemology—just like this has long beentaken as a good way of doing science. The second strategy is toexplain why it makes sense for we human beings tostrive forsome ideals, including the ideals that the two core Bayesian normspoint to, even though human beings cannot attain those ideals. Thethird strategy is to explain how the kind of idealization in questionactuallyempowers and facilitates the applications ofBayesian epistemology in science (including especiallyscientists’ use of Bayesian statistics). For more on thosereplies to the problem of idealization, seesection 6.

1.10 Concerns, or Encouragements, from Non-Bayesians

In the eyes of those immersed in the epistemology of all-or-nothingopinions such as believing or accepting propositions, Bayesians seemto say and care too little about many important and traditionalissues. Let me give some examples below.

First of all, the more traditional epistemologists would like to seeBayesians engage with varieties of skepticism. For example, there isCartesian skepticism, which is the view that we cannot knowwhether an external world, as we understand it through ourperceptions, exists. There is also thePyrrhonian skepticalworry that no belief can ever be justified because, once a belief isto be justified with a reason, the adduced reason is in need ofjustification as well, which kickstarts an infinite regress ofjustifications that can never be finished. Note that the aboveskeptical views are expressed in terms of knowledge and justification.So, the more traditional epistemologists would also like to hear whatBayesians have to say aboutknowledge andjustification, rather than just norms for credences.

Second, the more traditional philosophers of science would like to seeBayesians contribute to some classic debates, such as the one betweenscientific realism and anti-realism.Scientific realism is,roughly, the view that we have good reason to believe that our bestscientific theories are true, literally or approximately. But theanti-realists disagree. Some of them, such as theinstrumentalists, think that we only have good reason tobelieve that our best scientific theories are good tools for certainpurposes. Bayesians often compare the credences assigned to competingscientific theories, but one might like to see a comparison between,on the one hand, the credence that a certain theoryT is trueand, on the other hand, the credence thatT is a good tool forsuch and such purposes.

Last but not least, frequentists about statistical inference wouldurge that Bayesians also think about a certain epistemic virtue,reliability, rather than focus exclusively on coherence.Namely, they would like to see Bayesians take seriously the analysisand design of reliable inference methods—reliable in the senseof having a low objective, physical chance of making errors.

To be sure, Bayesian epistemology was not initially designed toaddress the concerns just expressed. But those concerns need not betaken as objections, but rather as encouragements to Bayesians toexplore new territories. In fact, Bayesians have begun suchexplorations in some of their more recent works, as you will see intheclosing section, 7.

The above finishes the introductory tutorial on Bayesian epistemology.The following sections, as well as many other encyclopedia entriescited above, elaborate on one or another more specific topic inBayesian epistemology. Indeed, the above tutorial only shows you whattopics there are and aims to help you jump to the sections below, orto the relevant entries, that interest you.

2. A Bit of Mathematical Formalism

To facilitate subsequent discussions, a bit of mathematical formalismis needed. Indeed, the two core Bayesian norms were only stated abovein a simple, finite setting (section 1.2), but there can be an infinity of possibilities under consideration.For example, think about this question: What’s the objective,physical chance for a carbon-14 atom to decay in 20 years? Everypossible chance in the unit interval $[0, 1]$ is a possibility towhich a credence can be assigned. So the two core Bayesian norms needto be stated in a more general way than above.

Let $\Omega$ be a set of possibilities that are mutually exclusiveand jointly exhaustive. There is no restriction on the size of$\Omega$; it can be finite or infinite. Let $\cal A$ be a set ofpropositions identified with some subsets of $\Omega$. Assume that$\cal A$ contains $\Omega$ and the empty set $\varnothing$, andis closed under the standard Boolean operations: conjunction(intersection), disjunction (union), and negation (complement). Thisclosure assumption means that, whenever $A$ and $B$ are in $\calA$, so are their intersection $A \cap B$, union $A \cup B$, andcomplement $\Omega \mcomplement A$, which are often written inlogical notation as conjunction $A \wedge B$, disjunction $A \veeB$, and negation $\neg A$. When $\cal A$ satisfies the assumptionjust stated, it is called analgebra of sets/propositions.^[2]

Let $\Cr$ be an assignment of credences to some propositions. Wewill often think of $\Cr(A)$ as denoting one’s credence inproposition $A$ and refer to $\Cr$ as one’scredencefunction orcredence assignment. Next, we need adefinition from probability theory:

Definition (Probability Measure). A credencefunction $\Cr(\wcdot)$ is said to beprobabilistic, alsocalled aprobability measure, if it is a real-valued functiondefined on an algebra ${\cal A}$ of propositions and satisfies thethree axioms of probability:
- (Non-Negativity) $\Cr(A) \ge 0$ for every$A$ in $\cal A$.
- (Normalization) $\Cr(\Omega) = 1$.
- (Finite Additivity) $\Cr(A \cup B) = \Cr(A) +\Cr(B)$ for any two incompatible propositions (i.e., disjoint sets)$A$ and $B$ in $\cal A$.

Now Probabilism can be stated as follows:

Probabilism (Standard Version). One’sassignment of credences at each time ought to be a probabilitymeasure.

When it is clear from the context that the credence assignment $\Cr$is assumed to be probabilistic, it is often written $\Pr$ or $P$.The process of conditionalization can be defined as follows:

Definition (Conditionalization). Suppose that$\Cr(E) \neq 0$. A (new) credence function $\Cr'(\wcdot)$ is saidto be obtained from (old) credence function $\Cr(\wcdot)$ byconditionalization on $E$ if, for each $X \in {\calA}$,
\[\Cr'(X) = \frac{\Cr(X\cap E)}{\Cr(E)}.\]

Conditionalization changes the credence in $X$ from $\Cr(X)$ to$\Cr'(X)$, which can be understood as involving two steps:

\[\Cr(X) \ovrightarrow{(i)}\Cr(X \cap E) \ovrightarrow{(ii)} \frac{\Cr(X\cap E)}{\Cr(E)} = \Cr'(X) .\]

Transition (i) corresponds to the zeroing step in the informalpresentation insection 1.2 of conditionalization; transition (ii), the rescaling step. Now thesecond norm can be stated as follows:

The Principle of Conditionalization (StandardVersion). One’s credences ought to change by and only byconditionalization on the new evidence received.

The two norms just stated reduce to the informal versions presented inthe tutorialsection 1.2 when $\Omega$ contains only finitely many possibilities and $\calA$ is the set of all subsets of $\Omega$.

Let $\Cr(X \mid E)$ denote one’s credence in $X$ on thesupposition of the truth of $E$ (whether or not one will actuallyreceive $E$ as new evidence); it is also called credence in $X$given $E$, or credence in $X$ conditional on $E$. So $\Cr(X\mid E)$ denotes aconditional credence, while $\Cr(X)$denotes anunconditional one. The connection between thosetwo kinds of credences is often expressed by

The Ratio Formula

\[\Cr(X\mid E) = \frac{\Cr(X \cap E)}{\Cr(E)} \quad\text{ if } \Cr(E) \neq 0.\]

It is debatable whether this formula should be construed as adefinition or as a normative constraint. See Hájek (2003) forsome objections to the definitional construal and for furtherdiscussion. $\Cr(X \mid E)$ is often taken as shorthand for thecredence in $X$ that results from conditionalization on $E$,assuming that the Ratio Formula holds.

Many applications of Bayesian epistemology make useBayes’theorem. It has different versions, of which two are particularlysimple:

Bayes’ Theorem (Simplest Version). Supposethat $\Cr$ is probabilistic and assigns nonzero credences to $H$and $E$, and that the Ratio Formula holds.^[3] Then we have:
\[ \Cr(H\mid E) = \frac{\Cr(E \mid H) \cdot \Cr(H)}{\Cr(E)} . \]

Bayes’ Theorem (Finite Version). Supposefurther that hypotheses $H_1, \ldots, H_N$ are mutually exclusiveand finite in number, and that each is assigned a nonzero credence andtheir disjunction is assigned credence 1 by $\Cr$. Then we have:
\[ \Cr(H_i\mid E) = \frac{\Cr(E \mid H_i) \cdot \Cr(H_i)}{\sum_{j=1}^{N} \Cr(E \mid H_j) \cdot \Cr(H_j)} . \]

This theorem is often useful for calculating credences that resultfrom conditionalization on evidence $E$, which are represented onthe left side of the formula. Indeed, this theorem is very useful andimportant in statistical applications of Bayesian epistemology (seesection 3.5 below). For more on the significance of this theorem, see the entryonBayes’ theorem. But this theorem is not essential to some other applications ofBayesian epistemology. Indeed, the case studies in the tutorialsection make no reference to Bayes’ theorem. As Earman (1992:ch. 1) points out in his presentation of Bayes’ (1763) seminalessay, Bayesian epistemology isBayesian not really becauseBayes’ theorem is used in a certain way, but becauseBayes’ essay already contains the core ideas of Bayesianepistemology: Probabilism and the Principle of Conditionalization.

Here are some introductory textbooks on Bayesian epistemology (andrelated topics) that include presentations of elementary probabilitytheory: Skyrms (1966 [2000]), Hacking (2001), Howson & Urbach(2006), Huber (2018), Weisberg (2019 [Other Internet Resources]), and Titelbaum (forthcoming).

3. Synchronic Norms (I): Requirements of Coherence

A coherence norm states how one’s opinions ought to fit togetheron pain of incoherence. Most Bayesians agree that the correctcoherence norms include at least Probabilism, but they disagree overwhich version of Probabilism is right. There is also the question ofwhether there are correct coherence norms that go beyond Probabilismand, if so, what they are. Those issues were only sketched in thetutorialsection 1.4. They will be detailed in this section.

To argue that a certain norm is not just correct but ought to befollowedon pain of incoherence, Bayesians traditionallyproceed by way of a Dutch Book argument (as presented in the tutorialsection 1.6). For the susceptibility to a Dutch Book is traditionally taken byBayesians to imply one’s personal incoherence. So, as you willsee below, the norms discussed in this section have all been defendedwith one or another type of Dutch Book argument, although it isdebatable whether some types are more plausible than others.

3.1 Versions of Probabilism

Probabilism is often stated as follows:

Probabilism (Standard Version). One’sassignment of credences ought to be probabilistic in this sense: it isa probability measure.

This norm implies that one should have a credence in a logical truth(indeed, a credence of 1) and that, when one has credences in somepropositions, one shouldalso have credences in theirconjunctions, disjunctions, and negations. So Probabilism in itsstandard version asks one to have credences in certain propositions.But that seems to be in tension with the fact that Probabilism isoften understood as acoherence norm. To see why, note thatcoherence is a matter of fitting things together nicely. So coherenceis supposed to put a constraint on the combinations of attitudes thatone may have,without saying that one must have an attitudetoward such and such propositions—contrary to the above versionof Probabilism. If so, the right version of Probabilism must be weakenough to allow the absence of some credences, also calledcredence gaps.

The above line of thought has led some Bayesians to develop and defenda weaker version of Probabilism (de Finetti 1970 [1974], Jeffrey 1983,Zynda 1996):

Probabilism (Extensibility Version).One’s assignment of credences ought to be probabilisticallyextensible in this sense: either it is already a probability measure,or it can be turned into a probability measure by assigning newcredences to some more propositions without changing the existingcredences.

It is the second disjunct that allows credence gaps. De Finetti (1970[1974: sec. 3]) also argues that, when the Dutch Book argument forProbabilism is carefully examined, it can be seen to support only theextensibility version rather than the standard one. His idea is toadopt a liberal conception of betting dispositions: one is permittedto lack any betting disposition about a proposition, which in turnpermits one to lack a credence in that proposition.

The above two versions of Probabilism are still similar in that theyboth imply that any credence ought to be sharp—being anindividual real number. But some Bayesians maintain that coherencedoes not require that much but allows credences to beunsharpin a certain sense. An even weaker version of Probabilism has beendeveloped accordingly, defended with a Dutch Book argument that workswith a more liberal conception of betting dispositions than mentionedabove (Smith 1961; Walley 1991: ch. 2 and 3). Seesupplement A for some non-technical details. Bayesians actually disagree overwhether coherence allows credences to be unsharp. For this debate, seethe survey by Mahtani (2019) and the entry onimprecise probabilities.

3.2 Countable Additivity

Probabilism, as stated insection 2, implies Finite Additivity, the norm that one’s credence in thedisjunction of two incompatible disjuncts ought to be equal to the sumof the credences in those two disjuncts. Finite Additivity can benaturally strengthened as follows:

Countable Additivity. It ought to be that, for anypropositions $A_1,$ $A_2,$…, $A_n,$… that aremutually exclusive, if one has credences in those propositions and intheir disjunction $\bigcup_{n=1}^{\infty} A_n$, then one’scredence function $\Cr$ satisfies the following formula:
\[\Cr\left( \bigcup_{n=1}^{\infty} A_n \right) = \sum_{n = 1}^{\infty} \Cr\left(A_n\right).\]

Countable Additivity has extensive applications, both in statisticsand in philosophy of science; for a concise summary and relevantreferences, see J. Williamson (1999: sec. 3).

Although Countable Additivity is a natural strengthening of FiniteAdditivity, the former is much more controversial. De Finetti (1970[1974]) proposes a counterexample:

Example (Infinite Lottery). There is a fairlottery with a countable infinity of tickets. Since it is fair, thereis one and only one winning ticket, and all tickets are equally likelyto win. For an agent taking all those for granted (i.e., with fullcredence), what should be her credence in the proposition $A_n$ thatthen-th ticket will win?

The answer seems to be 0. To see why, note that all those propositions$A_n$ should be assigned equal credences $c$, by the fairness ofthe lottery. Then it is not hard to show that, in order to satisfyProbabilism, a positive $c$ is too high and a negative $c$ is too low.^[4] So, by Probabilism, the only alternative is $c = 0$. But thisresult violates Countable Additivity: by the fairness of the lottery,the left side is

\[\Cr\left(\bigcup_{n = 1}^{\infty} A_n\right) = 1,\]

but the right side is

\[\sum_{n = 1}^{\infty} \Cr\left(A_n\right) = \sum_{n=1}^{\infty} c = 0.\]

De Finetti thus concludes that this is a counterexample to CountableAdditivity. For closely related worries about Countable Additivity,see Kelly (1996: ch. 13) and Seidenfeld (2001). Also see Bartha (2004:sec. 3) for discussions and further references.

Despite the above controversy, attempts have been made to argue forCountable Additivity, partly because of the interest in saving itsextensive applications. For example, J. Williamson (1999) defends theidea that there is a good Dutch Book argument for Countable Additivityeven though the Dutch Book involved has to contain a countableinfinity of bets and the agent involved has to be able to accept orreject that many bets. Easwaran (2013) provides further defense of theDutch Book argument for Countable Additivity (and another argument forit). The above two authors also argue that the Infinite Lottery Caseonly appears to be a counterexample to Countable Additivity and can beexplained away.

It is debatable whether we really need to defend Countable Additivityin order to save its extensive applications. Bartha (2004) thinks thatthe answer is negative. He argues that, even if Countable Additivityis abandoned due to the Infinite Lottery Case, this poses no seriousthreat to its extensive applications.

3.3 Regularity

A contingent proposition is true in some cases, while a logicalfalsehood is true in no cases at all. So perhaps the credence in theformer should always be greater than the credence in the latter, whichmust be 0. This line of thought motivates the following norm:

Regularity. It ought to be that, if one has acredence in a logically consistent proposition, it is greater than0.

Regularity has been defended with a Dutch Book argument—asomewhat nonstandard one. Kemeny (1955) and Shimony (1955) show thatany violation of Regularity opens the door to a nonstandard,weak Dutch Book, which is a set of bets that guarantees nogain but has a possible loss. In contrast, a standard Dutch Book has asure loss. This raises the question whether it is really so bad to bevulnerable to a weak Dutch Book.

One might object to Regularity on the ground that it is in conflictwith Conditionalization. To see the conflict, note thatconditionalization on a contingent proposition $E$ drops thecredence in another contingent proposition, $\neg E$, down to zero.But that violates Regularity. In reply, defenders of Regularity canreplace conditionalization by a generalization of it calledJeffrey Conditionalization, which need not drop any credencedown to zero. Jeffrey Conditionalization will be defined and discussedinsection 5.3.

There is a more serious objection to Regularity. Consider thefollowing case:

Example (Coin). An agent is interested in thebias of a certain coin—the objective, physical chancefor that coin to land heads when tossed. This agent’s credencesaredistributed uniformly over the possible biases of thecoin. This means that her credence in “the bias falls withininterval $[a, b]$” is equal to the length of the interval,$b-a$, provided that the interval is nested within $[0, 1]$. Nowthink about “the coin is fair”, which says that the biasis equal to 0.5, i.e., that the bias falls within the trivial interval$[0.5, 0.5]$. So “the coin is fair” is assigned credence$0.5 - 0.5$, which equals 0 and violates Regularity.

But there seems to be nothing incoherent in this agent’scredences.

One possible response is to insist on Regularity and hold that theagent in the Coin Case is actually incoherent in a subtle way. Namely,that agent’s credence in “the coin is fair” shouldnot be zero but should be aninfinitesimal—smaller thanany positive real number but still greater than zero (Lewis 1980). Onthis view, the fault lies not with Regularity but with the standardversion of Probabilism, which needs to be relaxed to permitinfinitesimal credences. For worries about this appeal toinfinitesimals, see Hájek (2012) and Easwaran (2014). For asurvey of infinitesimal credences/probabilities, see Wenmackers(2019).

The above response to the Coin Case implements a general strategy. Theidea is that some doxastic states are so nuanced that even realnumbers are too coarse-grained to distinguish them, so real-valuedcredences need to be supplemented withsomething else for abetter representation of one’s doxastic states. The aboveresponse proposes that the supplement beinfinitesimalcredences. A second response proposes, instead, that thesupplement becomparative probability, with a very differentresult: abandoning Regularity rather than saving it.

This second response can be developed as follows. While being assigneda higher numerical credence implies being taken as more probable,being assigned the same numerical credence does not really imply beingtaken as equally probable. That is, (real-valued) numerical credencesactually do not have enough structure to represent everything there isin a qualitative ordering of comparative probability, as Hájek(2003) suggests. So, in the Coin Case, the contingent proposition“the coin is fair” is assigned credence 0, the samecredence as a logical falsehood is assigned. But it does not mean thatthose two propositions, one contingent and one self-contradictory,should be taken as equally probable. Instead, the contingentproposition “the coin is fair” should still be taken asmore probable than a logical falsehood. That is, the following normstill holds:

Comparative Regularity. It ought to be that,whenever one has a judgment of comparative probability between acontingent proposition and a logical falsehood, the former is taken tobe more probable than the latter.

So, although the second response bites the bullet and abandonsRegularity (due to the Coin Case), it manages to settle on a variant,Comparative Regularity. But even Comparative Regularity can bechallenged: see T. Williamson (2007) for a putative counterexample.And see Haverkamp and Schulz (2012) for a reply in support ofComparative Regularity.

Note that the second response makes use of one’s ordering ofcomparative probability, which can be too nuanced to be fully capturedby real-valued credences. As it turns out, such an ordering can stillbe fully captured by real-valuedconditional credences (asexplained insupplement B), provided that it makes sense for a person to have a credence in aproposition conditional on azero-credence proposition. It isto this kind of conditional credence that I now turn.

3.4 Norms of Conditional Credences

In Bayesian epistemology, a doxastic state is standardly representedby a credence assignment $\Cr$, with conditional credencescharacterized by

The Ratio Formula

\[ \Cr(A\mid B) = \frac{\Cr(A \cap B)}{\Cr(B)}\quad \text{ if } \Cr(B) \neq 0.\]

The Ratio Formula might be taken to define conditional credences (onthe left) in terms of unconditional credences (on the right), or betaken as a normative constraint on those two kinds of mental stateswithout defining one by the other. See Hájek (2003) for someobjections to the definitional construal and for furtherdiscussion.

Whether the Ratio Formula is construed as a definition or a norm, itapplies only when the conditioning proposition $B$ is assigned anonzero credence: $\Cr(B) \neq 0$. But perhaps this qualification istoo restrictive:

Example (Coin, Continued). Conditional on“the coin is fair”, the agent has a 0.5 credence in“the coin will land heads the next time it istossed”—and rightly so. But this agent assigns azero credence in the conditioning proposition, “thecoin is fair”, as in the previous Coin Case.

This 0.5 conditional credence seems to make perfect sense, but iteludes the Ratio Formula. Worse, the above case is not rare: the aboveconditional credence is a credence in an event conditional on astatistical hypothesis, and such conditional credences, often calledlikelihoods, have been extensively employed in statisticalapplications of Bayesian epistemology (as will be explained insection 3.5).

There are three possible ways out. They differ in the importance theyattribute to the Ratio Formula as a stand-alone norm. So you canexpect a reformatory approach which takes it to be unimportant, aconservative one which retains its importance, and a middle waybetween the two.

On thereformatory approach, the Ratio Formula is no longerimportant and, instead, is derived as a mere consequence of somethingmore fundamental. While the standard Bayesian view takes norms ofunconditional credences to be fundamental and then uses the RatioFormula as a bridge to conditional credences, the reformatory approachreverses the direction, taking norms of conditional credences asfundamental. Following Popper (1959) and Rényi (1970), thisidea can be implemented with a version of Probabilism designeddirectly for conditional credences:

Probabilism (Conditional Version). It ought to bethat one’s assignment of conditional credences $\Cr( \wcdot\mid \wcdot)$ is a Popper-Rényi function over an algebra${\cal A}$ of propositions, namely, a function satisfying thefollowing axioms:
- (Probability) For any logically consistentproposition $A \in {\cal A}$ held fixed, $\Cr( \wcdot \mid A)$ isa probability measure on ${\cal A}$ with $\Cr( A \mid A) =1$.
- (Multiplication) For any propositions $A$,$B$, and $C$ in ${\cal A}$ such that $B \cap C$ is logicallyconsistent,
  \[\Cr(A\cap B \mid C) = \Cr(A \mid B \cap C) \cdot \Cr(B \mid C) .\]

This approach is often called the approach ofcoherent conditionalprobability, because it seeks to impose coherence constraintsdirectly on conditional credences without a detour throughunconditional credences. Once those constraints are in place, one maythen add a constraint—normative or definitional—onunconditional credences:

\[\Cr(A) = \Cr(A \mid \top),\]

where $\top$ is a logical truth. From the above we can derive theRatio Formula and the standard version of Probabilism. SeeHájek (2003) for a defense of this approach. A Dutch Bookargument for the conditional version of Probabilism is developed byStalnaker (1970).

In contrast to the reformatory nature of the above approach, thesecond one isconservative. On this approach, the RatioFormula is sufficient by itself as a norm (or definition) forconditional credences. It makes sense to have a credence conditionalon “the coin is fair” because one’s credence in thatconditioning proposition ought to be an infinitesimal rather thanzero. This approach may be called the approach ofinfinitesimals. It forms a natural package with theinfinitesimal approach to saving Regularity from the Coin Case, whichwas discussed insection 3.3.

Between the conservative and the reformatory, there is themiddle way, due to Kolmogorov (1933). The idea is to thinkabout the cases where the Ratio Formula applies, and then use them to“approximate” the cases where it does not apply. If thiscan be done, then although the Ratio Formula is not all there is tonorms for conditional credences, it comes close. To be more precise,when we try to conditionalize on a zero-credence proposition $B$, wecan approximate $B$ by a sequence of propositions $B_1,$$B_2,$… such that:

those propositions $B_1, B_2, \ldots$ are progressively morespecific (i.e., $B_i \supset B_{i+1}$),
they jointly say what $B$ says (i.e., $\bigcap_{i=1}^{\infty}B_i = B$).

In that case, it seems tempting to accept the norm or definition thatconditionalization on $B$ be approximated by successiveconditionalizations on $B_1, B_2, \ldots$, or in symbols:

\[\Cr(A \mid B) = \lim_{i \to \infty}\Cr(A \mid B_i),\]

where each term $\Cr(A \mid B_i)$ is governed by the Ratio Formulabecause $\Cr(B_i)$ is nonzero by design. An important consequence ofthis approach is that, when one chooses a different sequence ofpropositions to approximate $B$, the limit of conditionalizationsmight be different, and, hence, a credence conditional on $B$ is, orought to be, relativized to how one presents $B$ as the limit of asequence of approximating propositions. This relativization is oftenillustrated with what’s called theBorel-Kolmogorovparadox; see Rescorla (2015) for an accessible presentation anddiscussion. Once the mathematical details are refined, this approachbecomes what’s known as the theory ofregular conditional probability.^[5] A Dutch Book argument for this way of assigning conditional credencesis developed by Rescorla (2018).

For a critical comparison of those three approaches to conditionalcredences, see the survey by Easwaran (2019).

3.5 Chance-Credence Principles

Recall the Coin Case discussed above: one’s credence in“the coin will land heads the next time it is tossed”conditional on “the coin is fair” is equal to 0.5. This0.5 conditional credence seems to be the only permissible alternativeuntil the result of the next coin toss is observed. This examplesuggests a general norm, which connects chances to conditionalcredences:

The Principal Principle/Direct InferencePrinciple. Let $\Cr$ be one’s prior, i.e., the credenceassignment that one has at the beginning of an inquiry. Let $E$ bethe event that such and such things will happen at a certain futuretime. Let $A$ be a proposition that entails $\Ch(E) = c$, whichsays that the chance for $E$ to come out true is equal to $c$.Then one’s prior $\Cr$ ought to be such that $\Cr(E \mid A) =c$, if $A$ is an “ordinary” proposition in that it islogically equivalent to the conjunction of $\Ch(E) = c$ with an“admissible” proposition.

The if-clause refers to “admissible” propositions, whichare roughly propositions that give no more information about whetheror not $E$ is true than is already contained in $\Ch(E) = c$. Tosee why we need the qualification imposed by the if-clause, supposefor instance that the event $E$ is “the coin will land headsthe next time it is tossed”. If the conditioning proposition$A$ is “the coin is fair”, it is a paradigmatic exampleof an “ordinary” proposition. This reproduces the CoinCase, with the conditional credence being the chance 0.5.Alternatively, if the conditioning proposition $A$ is theconjunction of “the coin is fair” and $E$, then theconditional credence $\Cr(E \mid A)$ should be 1 rather than the 0.5chance of $E$ that $A$ entails. After all, to be given this $A$is to be given a lot of information, which entails $E$. So this caseis supposed to be ruled out by an account of “admissible”propositions. Lewis (1980) initiates a systematic quest for such anaccount, which has invited counterexamples and responses. See Joyce(2011: sec. 4.2) for a survey.

The Principal Principle has been defended with an argument based onconsiderations about the accuracies of credences (Pettigrew 2012), andwith a nonstandard Dutch Book argument (Pettigrew 2020a: sec.2.8).

The Principal Principle is important perhaps mainly because of itsextensive applications in Bayesian statistics, in which this principleis more often called the Direct Inference Principle. To illustrate,suppose that you are somehow certain that one of the following twohypotheses is true: $H_1 =$ “the coin has a bias 0.4”and $H_2 =$ “the coin has a bias 0.6”, which areparadigmatic examples of “ordinary” hypotheses. Then yourcredence in the first hypothesis $H_1$ given evidence $E$ that thecoin lands heads ought to be expressible as follows:^[6]

\[\begin{align} \Cr(H_1 \mid E) &= \frac{ \Cr(E \mid H_1) \cdot \Cr(H_1) }{ \sum_{i =1}^2 \Cr(E \mid H_i) \cdot \Cr(H_i) } &{\text{by Bayes' Theorem}\\ \text{(as stated in §2)}} \\ &= \frac{ 0.4 \cdot \Cr(H_1) }{ 0.4 \cdot \Cr(H_1) + 0.6 \cdot \Cr(H_2) } &{\text{by the Principal}\\ \text{Principle}} \end{align}\]

So Bayes’ Theorem works by expressing posterior credences interms of some prior credences $\Cr(H_i)$ and some prior conditionalcredences $\Cr(E \mid H_i)$. The latter, calledlikelihoods, aresubjective opinions, but they canbe replaced byobjective chances thanks to the PrincipalPrinciple. So this principle is often taken to be an important way toreduce some subjective factors in the Bayesian account of scientificinference. For discussions of other subjective factors, seesection 4.1.

Even though the Principal Principle has important, extensiveapplications in Bayesian statistics as just explained, de Finetti(1970 [1974]) argues that it is actually dispensable and thus need notbe accepted as a norm. To be more specific, he argues that thePrincipal Principle is dispensable in a way that changes little of theactual practice of Bayesian statistics. His argument relies on hisexchangeability theorem. See Gillies (2000: 69–82) fora non-technical introduction to this topic; also see Joyce (2011: sec.4.1) for a more advanced survey.

3.6 Reflection and Other Deference Principles

We have just discussed the Principal Principle, which in a sense asksone to defer to a kind of expert (Gaifman 1986): the chance of anevent $E$ can be understood as an expert at predicting whether $E$will come out true. So, conditional on that expert’s saying soand so about $E$, one’s opinion ought to defer to that expert.Construed that way, the Principal Principle is a kind ofdeferenceprinciple. There can be different deference principles, referringto different kinds of experts.

Here is another example of a deference principle, proposed by vanFraassen (1984):

The Reflection Principle. One’s credence atany time $t_1$ in a proposition $A$, conditional on theproposition that one’s future credence at $t_2$ $(> t_1)$in $A$ will be equal to $x$, ought to be equal to $x$; or putsymbolically:
\[\Cr_{t_1}( A \mid \Cr_{t_2}(A) = x ) = x.\]
More generally, it ought to be that
\[\Cr_{t_1}( A \mid \Cr_{t_2}(A) \in [x, x'] ) \in [x, x'].\]

Here, one’s future self is taken as an expert to which one oughtto defer. The Reflection Principle admits of a Dutch Book argument(van Fraassen 1984). There is another way to defend the ReflectionPrinciple: this synchronic norm is argued to follow from thesynchronic norm that one ought, at any time, to be fullycertain that one will follow thediachronic Principle ofConditionalization (as suggested by Weisberg’s 2007 modificationof van Fraassen’s 1995 argument).

The Reflection Principle has invited some putative counterexamples.Here is one, adapted from Talbott (1991):

Example (Dinner). Today is March 15, 1989.Someone is very confident that she is now having spaghetti for dinner.She is also very confident that, on March 15, 1990 (exactly one yearfrom today), she will have completely forgotten what she is having fordinner now.

So, this person’s current assignment of credences$\Cr_\textrm{1989}$ has the following properties, where $A$ is theproposition that she has spaghetti for dinner on March 15, 1989:

\[\begin{align} \Cr_\textrm{1989} \big( A \big) &= \text{high} \\ \Cr_\textrm{1989} \Big( \Cr_\textrm{1989+1}(A) \mbox{ is low} \Big) &= \text{high} . \end{align}\]

But conditionalization on a proposition with a high credence can onlyslightly change the credence assignment. For such a conditionalizationinvolves lowering just a small bit of credence down to zero and henceit only requires a slight rescaling, by a factor close to 1. So,assuming that $\Cr$ is a probability measure, we have:

\[ \Cr_\textrm{1989} \Big( A \Bigm\vert \Cr_\textrm{1989+1}(A) \mbox{ is low} \Big) = \text{still high} ,\]

which violates the Reflection Principle.

The Dinner Case serves as a putative counterexample to the ReflectionPrinciple by allowing one to suspect that one will lose some memories.So it allows one to have a specific kind ofepistemicself-doubt—to doubt one’s own ability to achieve orretain an epistemically favorable state. In fact, some are worriedthat the Reflection Principle is generally incompatible with epistemicself-doubt, which seems rational and permissible. For more on thisworry, see the entry onepistemic self-doubt.

4. Synchronic Norms (II): The Problem of the Priors

Much of what Bayesians have to say about confirmation and inductiveinference depends crucially on the norms that govern one’s priorcredences (the credences that one has at the beginning of an inquiry).But what are those norms? This is known as theproblem of thepriors. Some potential solutions were only sketched in thetutorialsection 1.5. They will be detailed in this section.

4.1 Subjective Bayesianism

Subjective Bayesianism is the view that every prior is permittedunless it fails to be coherent (de Finetti 1970 [1974]; Savage 1972;Jeffrey 1965; van Fraassen 1989: ch. 7). Holding that view as thecommon ground, subjective Bayesians often disagree over what coherencerequires (which was the topic of the precedingsection 3).

The most common worry for subjective Bayesianism is that, on thatview, anything goes. For example, under just Probabilism andRegularity, there is a prior that follows enumerative induction andthere also is a prior whose posterior never generalizes from data,defying enumerative induction (see Carnap 1955 for details, but seeFitelson 2006 for a concise presentation). Under just Probabilism andthe Principal Principle, there is a prior that follows Ockham’srazor in statistical model selection but there also is a prior thatdoes not (Forster 1995: sec. 3; Sober 2002: sec. 6).^[7] So, although subjective Bayesianism does not really say that anythinggoes, it seems to permit too much, failing to account for someimportant aspects of scientific objectivity—or so the worrygoes. Subjective Bayesians have replied with at least twostrategies.

Here is one: argue that, despite appearances, coherence alone captureseverything there is to scientific objectivity. For example, it mightbe argued that it is actually correct to permit a wide range ofpriors, for people come with different background opinions and itseems wrong—objectively wrong—to require all of them tochange to the same opinion at once. What ought to be the case is,rather, that people’s opinions be brought closer and closer toeach other as their shared evidence accumulates. This idea ofmerging-of-opinions as a kind of scientific objectivity canbe traced back to Peirce (1877), although he develops this idea forthe epistemology of all-or-nothing beliefs rather than credences. Somesubjective Bayesians propose to develop this Peircean idea in theframework of subjective Bayesianism: to have the ideal ofmerging-of-opinions be derived as a norm—derived solely fromcoherence norms. That is, they prove so-calledmerging-of-opinionstheorems (Blackwell & Dubins 1962; Gaifman & Snir 1982).Such a theorem states that, under such and such contingent initialconditions together with such and such coherence norms, two agentsmust becertain that their credences in the hypotheses underconsideration will merge with each otherin the long run asthe shared evidence accumulates indefinitely.

The above theorem is stated with two italicized parts, which are thetargets of some worries. The merging of the two agents’ opinionsmight not happen and is only believed with certainty to happen in thelong run. And the long run might be too long. There is another worry:the proof of such a theorem requires Countable Additivity as a norm ofcredences, which is controversial, as was discussed insection 3.2. See Earman (1992: ch. 6) for more on those worries.^[8] For a recent development of merging-of-opinions theorems and adefense of their use, see Huttegger (2015).

Whether or not merging-of-opinions theorems can capture the intendedkind of scientific objectivity, it is still debated whether there areother kinds of scientific objectivity that elude subjectiveBayesianism. For more on this issue, seesection 4.2 of the entry on scientific objectivity, Gelman & Hennig (2017) (including peer discussions), Sprenger(2018), and Sprenger & Hartmann (2019: ch. 11).

Here is a second strategy in defense of scientific objectivity forsubjective Bayesians: distance themselves from any substantive theoryof inductive inference and hold instead that Bayesian epistemology canbe construed as a kind of deductive logic. This view draws on someparallel features between deductive logic and Bayesian epistemology.First, the coherence of credences can be construed as an analogue ofthe logical consistency of propositions or all-or-nothing beliefs(Jeffrey 1983). Second, just as premises are inputs into a deductivereasoning process, prior credences are inputs into the process of aninquiry. And, just as the job of deductive logic is not to say whatpremises we should have except that they be logically consistent,Bayesian epistemology need not say what prior credences we should haveexcept that they be coherent (Howson 2000: 135–145). Call thisview thedeductive construal of Bayesian epistemology, forlack of a standard name.

Yet it might be questioned whether the above parallelism really worksin favor of subjective Bayesianism. Just as substantive theories ofinductive inferences have been developed with deductive logic as theirbasis, to take the parallelism seriously it seems that there shouldalso be a substantive account of inductive inferences with thedeductive construal of Bayesian epistemology as their basis. Indeed,the anti-subjectivists to be discussed below—objective Bayesiansand forward-looking Bayesians—all think that a substantiveaccount of inductive inferences is furnished by norms that go beyondthe consideration of coherence. It is to such a view that I turn now.But for more on subjective Bayesianism, see the survey by Joyce(2011).

4.2 Objective Bayesianism

Objective Bayesians contend that, in addition to coherence,there is another epistemic virtue or ideal that needs to be codifiedinto a norm for prior credences: freedom from bias and avoidance ofoverly strong opinions (Jeffreys 1939; Carnap 1945; Jaynes 1957, 1968;Rosenkrantz 1981; J. Williamson 2010). This view is often motivated bya case like this:

Example (Six-Faced Die). Suppose that there isa cubic die with six faces that look symmetric, and we are going totoss it. Suppose further that we have no other idea about this die.Now, what should our credence be that the die will come up 6?

An intuitive answer is $1/6$, for it seems that we ought todistribute our credences evenly, with an equal credence, $1/6$, ineach of the six possible outcomes. While subjective Bayesians wouldonly say that wemay do so, objective Bayesians would makethe stronger claim that weought to do so. More generally,objective Bayesians are sympathetic to this norm:

The Principle of Indifference. Aperson’s credences in any two propositions should be equal ifher total evidence no more supports one than the other (theevidential symmetry version), or if she has no sufficientreason to have a higher credence in one than in the other (theinsufficient reason version).

A standard worry about the Indifference Principle comes fromBertrand’s paradox. Here is a simplified version(adapted from van Fraassen 1989):

Example (Square). Suppose that there is asquare and that we know for sure that its side length is between 1 and4 centimeters. Suppose further that we have no other idea about thatsquare. Now, how confident should we be that the square has a sidelength between 1 and 2 centimeters?

Now, have a look at the two groups of propositions listed in the tablebelow. The left group (1)–(3) focuses on possible side lengthsand divides up possibilities by 1-cm-long intervals; the right group$(1')$–$(15')$ focuses on possible areas instead:

Partition ByLength	Partition ByArea
(1) The side length is 1 to 2 cm.	$(1')$ The area is 1 to 2cm².
(2) The side length is 2 to 3 cm.	$(2')$ The area is 2 to 3cm².
(3) The side length is 3 to 4 cm.	$(3')$ The area is 3 to 4cm².
	$\;\;\vdots$
	$(15')$ The area is 15 to 16cm²

The Indifference Principle seems ask us to assign a $1/3$ credenceto each proposition in the left group $(1)$–$(3)$ and,simultaneously, assign $1/15$ to each one in the right group$(1')$–$(15')$. If so, it asks us to assign unequalcredences to equivalent propositions: $1/3$ to $(1)$, and $3/15$to the disjunction $(1') \!\vee (2') \!\vee (3')$. That violatesProbabilism.

In reply, objective Bayesians may reply that Bertrand’s paradoxprovides no conclusive reason against the Indifference Principle andperhaps the fault lies elsewhere. Following White (2010), let’sthink about how the Indifference Principle works: it outputs anormative recommendation for credence assignment only when it receivesone or anotherinput, which is a judgement about insufficientreason or evidential symmetry. Indeed, Bertrand’s paradox has tobe generated by at least two inputs, such as, first, thelack-of-evidence judgement about the left group in the above tableand, second, that about the right group. So perhaps the fault lies notwith the Indifference Principle but with one of the twoinputs—after all, garbage in, garbage out. White (2010)substantiates the above idea with an argument to this effect: at leastone of the two inputs in Bertrand’s paradox must be mistaken,because they already contradict each other even when we only assumecertain weak, plausible principles that have nothing to do withcredences and concern just the evidential support relation.

There still remains the task of developing a systematic account toguide one’s judgments of evidential symmetry (or insufficientreason) before those judgments are passed as inputs to theIndifference Principle. An important source of inspiration has beenthe symmetry in the Six-Faced Die Case: it is a kind ofphysical symmetry due to the cubic shape of the die; it isalso a kind ofpermutation symmetry because nothing essentialchanges when the six faces of the die are relabeled. Those two aspectsof the symmetry—physical and permutational—are extended bytwo influential approaches to the Indifference Principle,respectively, which are presented in turn below.

The first approach to the Indifference Principle looks for a widerrange ofphysical symmetries, including especially thesymmetries associated with a change of coordinate or unit. Thisapproach, developed by Jeffreys (1946) and Jaynes (1968, 1973), yieldsa consistent, somewhat surprising answer 1/2 (rather than 1/3 or 1/15)to the question in the Square Case. Seesupplement C for some non-technical details.

The second approach to the Indifference Principle focuses onpermutation symmetries and proposes to look for those not ina physical system but in thelanguage in use. This approachis due to Carnap (1945, 1955). He maintains, for example, that twosentences ought to be assigned equal prior credences if one differsfrom the other only by a permutation of the names in use. AlthoughCarnap says little about the Square Case, he has much to say about howhis approach to the Indifference Principle helps to justifyenumerative induction; see the survey by Fitelson (2006). So objectiveBayesianism is often regarded as a substantive account of inductiveinference, while many subjective Bayesians often take their view as aquantitative analogue of deductive logic (as presented insection 4.1). For refinement of Carnap’s approach, see Maher (2004). The mostcommon worry for Carnap’s approach is that it renders thenormative import of the Indifference Principle too sensitive to thechoice of a language; for a reply, see J. Williamson (2010: chap. 9).For more criticisms, see Kelly & Glymour (2004).

The Indifference Principle has been challenged for another reason.This principle is often understood to dictate equalreal-valued credences in cases of ignorance, but there is theworry that sometimes we are too ignorant to be justified in havingsharp, real-valued credences, as suggested by this case (Keynes 1921:ch. 4):

Example (Two Urns). Suppose that there are twourns,a andb. Urna contains 10 balls. Exactlyhalf of those are white; the other half, black. Urnb contains10 balls, each of which is either black or white, but we have no ideaabout the white-to-black ratio. Those two urns are each shaken well. Aball is to be drawn from each. What should our credences be in thefollowing propositions?
- (A) The ball from urna is white.
- (B) The ball from urnb is white.

By the Principle of Indifference, the answers seems to be 0.5 and 0.5,respectively. If so, there should be equal credences (namely 0.5) inA and inB. But this result sounds wrong to Keynes. Hethinks that, compared with urna, we have much less backgroundinformation about urnb, and that this severe lack ofbackground information should be reflected in the difference betweenthe doxastic attitudes toward propositionsA andB—a difference that the Principle of Indifference failsto make. If so, what is the difference? It is relativelyuncontroversial that the credence inA should be 0.5, being theratio of the white balls in urna (perhaps thanks to thePrincipal Principle). On the other hand, some Bayesians (Keynes 1921;Joyce 2005) argue that the credence inB does not have to be anindividual real number but, instead, is at least permitted to beunsharp, being the interval $[0, 1]$, which covers all the possiblewhite-to-black ratios under consideration. This is only one motivationfor an interval account ofunsharp credences; for anothermotivation, seesupplement A.

In reply to the Two Urns Case, objective Bayesians have defended oneor another version of the Indifference Principle. White (2010) does itwhile maintaining that credences ought to be sharp. Weatherson (2007:sec. 4) defends a version that allows credences to be unsharp. Eva(2019) defends a version that governs comparative probabilities ratherthan numerical credences. For more on this debate, see the survey byMahtani (2019) and the entry onimprecise probabilities.

The Principle of Indifference appears unhelpful when one has hadsubstantive reason or evidence against some assignments of credences(making the principle inapplicable with a false if-clause). Thestandard remedy appeals to a generalization of the IndifferencePrinciple, calledthe Principle of Maximum Entropy (Jaynes1968); for more on this, seesupplement D.

The above has only mentioned the versions of objective Bayesianismthat are more well-known in philosophy. There are other versions,developed and discussed mostly by statisticians. For a survey, seeKass & Wasserman (1996) and Berger (2006).

4.3 Forward-Looking Bayesianism

Some Bayesians propose that some norms for priors can be obtained bylooking into possible futures, with two steps (Good 1976):

Step I (Think Ahead). Develop a normativeconstraintC on the posteriors in some possible futures inwhich new evidence is acquired.
Step II (Solve Backwards). Require one’spriors to be such that, after conditionalization on new evidence, itsposterior must satisfyC.

For lack of a standard name, this approach may be calledforward-looking Bayesianism. This name is used here as anumbrella term to cover different possible implementations, of whichtwo are presented below.

Here is one implementation. It might be held that one ought to favor ahypothesis if it explains the available evidence better than any othercompeting hypotheses do. This view is calledinference to the bestexplanation (IBE) if construed as a method for theory choice, asoriginally developed in the epistemology of all-or-nothing beliefs(Harman 1986). It can be carried over to Bayesian epistemology asfollows:

Explanationist Bayesianism (PreliminaryVersion). One’s prior ought to be such that, given eachbody of evidence under consideration, a hypothesis that explains theevidence better has a higher posterior.

What’s stated here is only a preliminary version. Moresophisticated versions are developed by Lipton (2004: ch. 7) andWeisberg (2009a). This view is resisted by some Bayesians to varyingdegrees. van Fraassen (1989: ch. 7) argues that IBE should be rejectedbecause it is in tension with the two core Bayesian norms. Okasha(2000) argues that IBE only serves as a good heuristic for guidingone’s credence change. Henderson (2014) argues that IBE need notbe assumed to guide one’s credence change because it can bejustified by little more than the two core Bayesian norms. For more onIBE, see the entry onabduction, in which sections 3.1 and 4 discuss explanationist Bayesianism.

Here is another implementation of forward-looking Bayesianism. Itmight be thought that, although a scientific method for theory choiceis subject to error due to its inductive nature, it is supposed to beable, in a sense, to correct itself. This view is calledtheself-corrective thesis, originally developed in the epistemologyof all-or-nothing beliefs by Peirce (1903) and Reichenbach (1938: sec.38–40). But it can be carried over to Bayesian epistemology asfollows:

Self-Correctionist Bayesianism (PreliminaryVersion). One’s prior ought, if possible, to have at leastthe following self-corrective property in every possible state of theworld under consideration: one’s posterior credence in the truehypothesis under consideration would eventually become high and stayso if the evidence were to accumulate indefinitely.

An early version of this view is developed by Freedman (1963) instatistics; see Wasserman (1998: sec. 1–3) for a minimallytechnical overview. The self-corrective property concerns the longrun, so it invites the standard, Keynesian worry that the long runmight be too long. For replies, see Diaconis & Freedman (1986b:pp. 63–64) and Kelly (2000: sec. 7). A related worry is that along-run norm puts no constraint on what matters, namely, our doxasticstates in the short run (Carnap 1945). A possible reply is that theself-corrective property is only a minimum qualification ofpermissible priors and can be conjoined with other norms for credencesto generate a significant constraint on priors. To substantiate thatreply, it has been argued that such a constraint on priors is actuallystronger than what the rival Bayesians have to offer in some importantcases of statistical inference (Diaconis & Freedman 1986a) andenumerative induction (Lin forthcoming).

The above two versions of forward-looking Bayesianism both encourageBayesians to do this: assimilate some ideas (such as IBE orself-correction) that have long been taken seriously in somenon-Bayesian traditions of epistemology. Forward-looking Bayesianismseems to be a convenient template for doing that.

4.4 Connection to the Uniqueness Debate

The above approaches to the problem of the priors are mostly developedwith this question in mind:

The Question of Norms. What are the correctnorms that we can articulate to govern prior credences?

The interest in this question leads naturally to a different butclosely related question. Imagine that you are unsympathetic tosubjective Bayesianism. Then you might try to add one norm afteranother to narrow down the candidate pool for the permissible priors,and you might be wondering what this process might end up with. Thisraises a more abstract question:

The Question of Uniqueness. Given eachpossible body of evidence, is there exactly one permissible credenceassignment or doxastic state (whether or not we can articulate normsto single out that state)?

Impermissive Bayesianism is the view that says“yes”;permissive Bayesianism says“no”. The question of uniqueness is often addressed in away that is somewhat orthogonal to the question of norms, as issuggested by the ‘whether-or-not’ clause in theparentheses. Moreover, the uniqueness question is often debated in abroader context that considers not just credences but all possibledoxastic states, thus going beyond Bayesian epistemology. Readersinterested in the uniqueness question are referred to the survey byKopec and Titelbaum (2016).

Let me close this section with some clarifications. The two terms‘objective Bayesianism’ and ‘impermissiveBayesianism’ are sometimes used interchangeably. But those twoterms are used in the present entry to distinguish two differentviews, and neither implies the other. For example, many prominentobjective Bayesians such as Carnap (1955), Jaynes (1968), and J.Williamson (2010) are not committed to impermissivism, even thoughsome objective Bayesians tend to be sympathetic to impermissivism. Forelaboration on the point just made, seesupplement E.

5. Issues about Diachronic Norms

The Principle of Conditionalization has been challenged with severalputative counterexamples. This section will examine some of the mostinfluential ones. We will see that, to save that principle, someBayesians have tried to refine it into one or another version. Anumber of versions have been systematically compared in papers such asthose of Meacham (2015, 2016), Pettigrew (2020b), and Rescorla (2021),while the emphasis below will be centered on the proposedcounterexamples.

5.1 Old Evidence

Let’s start with the problem of old evidence, which waspresented above (in the tutorialsection 1.8) but is reproduced below for ease of reference:

Example (Mercury). It is 1915. Einstein hasjust developed a new theory, General Relativity. He assesses the newtheory with respect to some old data that have been known for at leastfifty years: the anomalous rate of the advance of Mercury’sperihelion (which is the point on Mercury’s orbit that isclosest to the Sun). After some derivations and calculations, Einsteinsoon recognizes that his new theory entails the old data about theadvance of Mercury’s perihelion, while the Newtonian theory doesnot. Now, Einstein increases his credence in his new theory, andrightly so.

There appears to be no change in the body of Einstein’s evidencewhen he is simply doing some derivations and calculations. But thelimiting case of no new evidence seems to be just the case inwhich the new evidenceE is trivial, being a logical truth,ruling out no possibilities. Now, conditionalization on new evidenceE as a logical truth changes no credence; but Einstein changeshis credences nonetheless—and rightly so. This is calledtheproblem of old evidence, formulated as a counterexample to thePrinciple of Conditionalization.

To save the Principle of Conditionalization, a standard reply is tonote that Einstein seems to discover something new, a logicalfact:

$(E_\textrm{logical})$ The new theory, together with such andsuch auxiliary hypotheses, logically implies such and such oldevidence.

The hope is that, once this proposition has a less-than-certaincredence, Einstein’s credence change can then be explained andjustified as a result of conditionalization on this proposition(Garber 1983, Jeffrey 1983, and Niiniluoto 1983). There are fourworries about this approach.

An initial worry is that the discovery of the logical fact$E_\textrm{logical}$ does not sound like adding anything to the bodyof Einstein’s evidence but seemsonly to make clear theevidential relation between the new theory and the existing,unaugmented body of evidence. If so, there is no new evidence afterall. This worry might be addressed by providing a modified version ofthe Conditionalization Principle, according to which the thing to beconditionalized on is not exactly what one acquires as new evidencebut, rather, what onelearns. Indeed, it seems to soundnatural to say that Einstein learns something nontrivial from hisderivations. For more on the difference between learning and acquiringevidence, see Maher (1992: secs 2.1 and 2.3). So this approach to theproblem of old evidence is often calledlogical learning.

A second worry for the logical learning approach points to an internaltension: On the one hand, this approach has to work by permitting aless-than-certain credence in a logical fact such as$E_\textrm{logical}$, and that amounts to permitting one to make acertain kind of logical error. On the other hand, this approach hasbeen developed on the assumption of Probabilism, which seems torequire that one be logically omniscient and make no logical error (asmentioned in the tutorialsection 1.9). van Fraassen (1988) argues that these two aspects of the logicallearning approach contradict each other under some weakassumptions.

A third worry is that the logical learning approach depends for itssuccess on certain questionable assumptions about prior credences. Forcriticisms of those assumptions as well as possible improvements, seeSprenger (2015), Hartmann & Fitelson (2015), and Eva &Hartmann (2020).

There is a fourth worry, which deserves a subsection of its own.

5.2 New Theory

The logical learning approach to the problem of old evidence invitesanother worry. It seems to fail to address a variant of the MercuryCase, due to Earman (1992: sec. 5.5):

Example (Physics Student). A physics studentjust started studying Einstein’s theory of general relativity.Like most physics students, the first thing she learns about thetheory, even before hearing any details of the theory itself, is thelogical fact $E_\textrm{logical}$ as formulated above. Afterlearning that, this student forms aninitial credence 1 in$E_\textrm{logical}$, and an initial credence in the new,Einsteinian theory. She also lowers her credence in the old, Newtoniantheory.

The student’sformation of a new, initial credence inthe new theory seems to pose a relatively little threat to thePrinciple of Conditionalization, which is most naturally construed asa norm that governs, not credence formation, but credence change. Sothe more serious problem lies in the student’schangeof her credence in the old theory. If this credence drop reallyresults from conditionalization on what was just learned,$E_\textrm{logical}$, then the credence in $E_\textrm{logical}$must be boosted to 1 from somewhere below 1, which unfortunately neverhappens. So it seems that the student’s credence drop violatesthe Principle of Conditionalization and rightly so, which is known asthe problem of new theory. The following presents two replystrategies for Bayesians.

One reply strategy is to qualify the Conditionalization Principle andmake it weaker in order to avoid counterexamples. The following is oneway to implement this strategy (seesupplement F for another one):

The Principle of Conditionalization (Plan/RuleVersion). It ought to be that, if one has a plan (or follows arule) for changing credences in the case of learningE, thenthe plan (or rule) is to conditionalize onE.

Note how this version is immune from the Physics Student Case: what islearned, $E_\textrm{logical}$, is something entirely new to thestudent, so the student simply did not have in mind a plan forresponding to $E_\textrm{logical}$—so the if-clause is notsatisfied. The Bayesians who adopt this version, such as van Fraassen(1989: ch. 7), often add that one isnot required to have aplan for responding to any particular piece of new evidence.

The plan version is independently motivated. Note that this versionputs a normative constraint on theplan that one hasateach time when one has a plan, whereas the standard versionconstrains theact of credence changeacross differenttimes. So the plan version is different from the standard, actversion. But it turns out to be the former, rather then the latter,that is supported by the major existing arguments for the Principle ofConditionalization. See, for example, the Dutch Book argument by Lewis(1999), the expected accuracy argument by Greaves & Wallace(2006), and the accuracy dominance argument by Briggs & Pettigrew(2020).

While the plan version of the Conditionalization Principle is weakenough to avoid the Physics Student counterexample, it might beworried that it is too weak. There are actually two worries here. Thefirst worry is that the plan version is too weak because it leavesopen an important question: Even if one’s plan for credencechange is always a plan to conditionalize on new evidence, should oneactually follow such a plan whenever new evidence is acquired? Fordiscussions of this issue, see Levi (1980: ch. 4), van Fraassen (1989:ch. 7), and Titelbaum (2013a: parts III and IV). (Terminological note:instead of ‘plan’, Levi uses ‘confirmationalcommitment’ and van Fraassen uses ‘rule’.) Thesecond worry is that the plan version is too weak because it onlyavoids the problem of new theory, without giving a positive account asto why the student’s credence in the old theory ought todrop.

A positive account is promised by the next strategy for solving theproblem of new theory. It operates with a series of ideas. The firstidea is that, typically, a person only considers possibilities thatare not jointly exhaustive, and she only has credencesconditional on the setC of the consideredpossibilities—lacking an unconditional credence inC(Shimony 1970; Salmon 1990). This deviates from the standard Bayesianview in allowing two things: credence gaps (section 3.1), and primitive conditional credences (section 3.4). The second idea is that the setC of the consideredpossibilities might shrink or expand in time. It might shrink becausesome of those possibilities are ruled out by new evidence, or it mightexpand because a new possibility—a new theory—is takeninto consideration. The third and last idea is a diachronic norm(sketched by Shimony 1970 and Salmon 1990, developed in detail byWenmackers & Romeijn 2016):

The Principle of Generalized Conditionalization(Considered Possibilities Version). It ought to be that, if twopossibilities are under consideration at an earlier time and remain soat a later time, then their credence ratio be preserved across thosetwo times.

Here, a credence ratio has to be understood in such a way that it canexist without any unconditional credence. To see how this is possible,suppose for simplicity that an agent starts with two old theories asthe only possibilities under consideration, $\mathsf{old}_1$ and$\mathsf{old}_2$, with a credence ratio $1:2$ but without anyunconditional credence. This can be understood to mean that, while theagent lacks an unconditional credence in the set $\{\mathsf{old}_1 ,\mathsf{old}_2\}$, she still has a conditional credence$\frac{1}{1+2}$ in $\mathsf{old}_1$ given that set. Now, supposethat this agent then thinks of a new theory: $\mathsf{new}$. Then,by the diachronic norm stated above, the credence ratio among$\mathsf{old}_1$, $\mathsf{old}_2$, $\mathsf{new}$ should now be$1:2:x$. Notice the change of this agent’s conditionalcredence in $\mathsf{old}_1$ given thevarying set of theconsidered possibilities: it drops from $\frac{1}{1+2}$ down to$\frac{1}{1+2+x}$, provided that $x>0$. Wenmackers &Romeijn (2016) argues that this is why there appears to be a drop inthe student’s credence in the old theory—it is actually adrop in a conditional credence given the varying set of the consideredpossibilities.

The above account invites a worry from the perspective of rationalchoice theory. According to the standard construal of Bayesiandecision theory, the kind of doxastic state that ought to enterdecision-making isunconditional credence rather thanconditional credence. So Earman (1992: sec. 7.3) is led to think thatwhat we really need is an epistemology forunconditionalcredence, which the above account fails to provide. A possible replyis anticipated by some Bayesian decision theorists, such as Savage(1972: sec. 5.5) and Harsanyi (1985). They argue that, when making adecision, we often only have conditional credences—conditionalon a simplifying assumption that makes the decision problem inquestion manageable. For other Bayesian decision theorists who followSavage and Harsanyi, see the references in Joyce (1999: sec. 2.6, 4.2,5.5 and 7.1). For more on rational choice theory, see the entry ondecision theory and the entry onnormative theories of rational choice: expected utility.

5.3 Uncertain Learning

When we change our credences, the Principle of Conditionalizationrequires us to raise the credence in some proposition, such as thecredence in the new evidence, all the way to 1. But it seems that weoften have credence changes that do not accompany such as a radicalrise to certainty, as witnessed by the following case:

Example (Mudrunner). A gambler is veryconfident that a certain racehorse, called Mudrunner, performsexceptionally well on muddy courses. A look at the extremely cloudysky has an immediate effect on this gambler’s opinion: anincrease in her credence in the proposition $(\textsf{muddy})$ thatthe course will be muddy—an increasewithout reachingcertainty. Then this gambler raises her credence in the hypothesis$(\textsf{win})$ that Mudrunner will win the race, but nothingbecomes fully certain. (Jeffrey 1965 [1983: sec. 11.3])

Conditionalization is too inflexible to accommodate this case.

Jeffrey proposes a now-standard solution that replacesconditionalization by a more flexible process for credence change,calledJeffrey conditionalization. Recall thatconditionalization has a defining feature: it preserves the credenceratios of the possibilities inside new evidenceE while thecredence inE is raised all the way to 1. Jeffreyconditionalization does something similar: it preserves the samecredence ratioswithout having to raise any credence to 1,and also preserves someother credence ratios, i.e., thecredence ratios of the possibilities outsideE. A simpleversion of Jeffrey’s norm can be stated informally as follows(in the style of the tutorialsection 1.2):

The Principle of Jeffrey Conditionalization (SimplifiedVersion). It ought to be that, if the direct experiential impacton one’s credences causes the credence inE to rise to areal numbere (which might be less than 1), then one’scredences are changed as follows:
- For the possibilities insideE, rescale their credencesupward by a common factor so that they sum toe; for thepossibilities outsideE, rescale their credences downward by acommon factor so that they sum to $1-e$ (to obey the rule ofSum-to-One).
- Reset the credence in each propositionH by adding up thenew credences in the possibilities insideH (to obey the ruleof Additivity).

This reduces to standard conditionalization in the special case that$e = 1$. The above formulation is quite simplified; seesupplement G for a general statement. This principle has been defended with aDutch Book argument; see Armendt (1980) and Skyrms (1984) fordiscussions.

Jeffrey conditionalization is flexible enough to accommodate theMudrunner Case. Suppose that the immediate effect of thegambler’s sky-looking experience is to raise the credence in$E$, i.e. $\Cr(\mathsf{muddy})$. One feature of Jeffreyconditionalization is that, since certain credence ratios are requiredto be held constant, one has to hold constant the conditionalcredences given $E$ and also those given $\neg E$, such as$\Cr(\mathsf{win} \mid \mathsf{muddy})$ and $\Cr(\mathsf{win} \mid\neg\mathsf{muddy})$. The credences mentioned above can be used toexpress $\Cr(\mathsf{win})$ as follows (thanks to Probabilism andthe Ratio Formula):

\[\begin{multline} \Cr(\mathsf{win}) = \underbrace{\Cr(\mathsf{win} \mid \mathsf{muddy})}_\textrm{high, held constant} \wcdot \underbrace{\Cr(\mathsf{muddy})}_\textrm{raised} \\ {} + \underbrace{\Cr(\mathsf{win} \mid \neg\mathsf{muddy})}_\textrm{low, held constant} \wcdot \underbrace{\Cr(\neg\mathsf{muddy})}_\textrm{lowered}. \end{multline}\]

It seems natural to suppose that the first conditional credence ishigh and the second is low, by the description of the Mudrunner Case.The annotations in the above equation imply that $\Cr(\mathsf{win})$must go up. This is how Jeffrey conditionalization accommodates theMudrunner Case.

Although Jeffrey conditionalization is more flexible thanconditionalization, there is the worry that it is still too inflexibledue to something it inherits from conditionalization: the preservationof certain credence ratios or conditional credences (Bacchus, Kyburg,& Thalos 1990; Weisberg 2009b). Here is an example due to Weisberg(2009b: sec. 5):

Example (Red Jelly Bean). An agent with a prior$\Cr_\textrm{old}$ has a look at a jelly bean. The reddishappearance of that jelly bean has only one immediate effect on thisagent’s credences: an increased credence in the propositionthat
$(\textsf{red})$
there is a red jelly bean.
Then this agent comes to have a posterior $\Cr_\textrm{new}$. Ifthis agent later learns that
$(\textsf{tricky})$
the lighting is tricky,
her credence in the redness of the jelly bean will drop. So,
($a$)
$\Cr_\textrm{new}( \textsf{red} \mid \textsf{tricky} ) <\Cr_\textrm{new}( \textsf{red} )$.
But if, instead, the tricky lighting had been learnedbeforethe look at the jelly bean, it would not have changed the credence inthe jelly bean’s redness; that is:
($b$)
$\Cr_\textrm{old}( \textsf{red} \mid \textsf{tricky} ) =\Cr_\textrm{old}( \textsf{red} ).$

Yet it can be proved (with elementary probability theory) that$\Cr_\textrm{new}$ cannot be obtained from $\Cr_\textrm{old}$ by aJeffrey conditionalization on $\textsf{red}$ (assuming the twoconditions $(a)$ and $(b)$ in the above case, the Ratio Formula,and that $\Cr_\textrm{old}$ is probabilistic). Seesupplement H for a sketch of proof.

The above example is used by Weisberg (2009b) not just to argueagainst the Principle of Jeffrey Conditionalization, but also toillustrate a more general point: that principle is in tension with aninfluential thesis calledconfirmational holism, mostfamously defended by Duhem (1906) and Quine (1951). Confirmationalholism says roughly that how one should revise one’s beliefsdepends on a good deal of one’s background opinions—suchas the opinions about the quality of the lighting, the reliability ofone’s vision, the details of one’s experimental setup(which are conjoined with a tested scientific theory to predictexperimental outcomes). In reply, Konek (forthcoming) develops anddefends an even more flexible version of conditionalization, flexibleenough to be compatible with confirmational holism. For more onconfirmational holism, see the entry onunderdetermination of scientific theory and the survey by Ivanova (2021).

For a more detailed discussion of Jeffrey conditionalization, see thesurveys by Joyce (2011: sec. 3.2 and 3.3) and Weisberg (2011: sec. 3.4and 3.5).

5.4 Memory Loss

Conditionalization in the standard version preserves certainties,which fails to accommodate cases of memory loss (Talbott 1991):

Example (Dinner). At 6:30 PM on March 15,1989, Bill is certain that he is having spaghetti for dinner thatnight. But by March 15 of the next year, Bill has completely forgottenwhat he had for dinner one year ago.

There are even putative counterexamples that appear to beworse—with an agent who faces only the danger of memory lossrather than actual memory loss. Here is one such example (Arntzenius2003):

Example (Shangri-La). A traveler has reached afork in the road to Shangri-La. The guardians will flip a fair coin todetermine her path. If it comes up heads, she will travel the path bythe Mountains and correctly remember that all along. If instead itcomes up tails, she will travel by the Sea—with her memoryaltered upon reaching Shangri-La so that she will incorrectly rememberhaving traveled the path by the Mountains. So, either way, once inShangri-La the traveler will remember having traveled the path by theMountains. The guardians explain this entire arrangement to thetraveler, who believes those words with certainty. It turns out thatthe coin comes up heads. So the traveler travels the path by theMountains and has credence 1 that she does. But once she reachesShangri-La and recalls the guardians’ words, that credencesuddenly drops from 1 down to 0.5.

That credence drop violates the Principle of Conditionalization, andall that happens without any actual loss of memory.

It may be replied that conditionalization can be plausibly generalizedto accommodate the above case. Here is an attempt made by Titelbaum(2013a: ch. 6), who develops an idea that can be traced back to Levi(1980: sec. 4.3):

The Principle of Generalized Conditionalization(Certainties Version). It ought to be that, if two consideredpossibilities each entail one’s certainties at an earlier timeand continue to do so at a later time, then their credence ratio arepreserved across those two times.

This norm allows the set of one’s certainties to expand orshrink, while incorporating the core idea of conditionalization:preservation of credence ratios. To see how this norm accommodates theShangri-La Case, assume for simplicity that the traveler starts at theinitial time with a set of certainties, which expands upon seeing thecoin toss result at a later time, but shrinks back to theoriginal set of certainties upon reaching Shangri-La at thefinal time. Note that there is no change in one’s certaintiesacross the initial time and the final time. So, by the above norm,one’s credences at the final time (upon reaching Shangri-La)should be identical to those at the initial time (the start of thetrip). In particular, one’s final credence in traveling the pathby the Mountains should be the same as the initial credence, which is0.5. For more on the attempts to save conditionalization from cases ofactual or potential memory loss, see Meacham (2010), Moss (2012), andTitelbaum (2013a: ch. 6 and 7).

The Principle of Generalized Conditionalization, as stated above,might be thought to be an incomplete diachronic norm because it leavesopen the question of how one’s certainties ought to change.Early attempts at a positive answer are due to Harper (1976, 1978) andLevi (1980: ch. 1–4). Their ideas are developed independently ofthe issue of memory loss, but are motivated by the scenarios in whichan agent finds a need to revise or even retract what she used to taketo be her evidence. Although Harper’s and Levi’sapproaches are not identical, they share the common idea thatone’s certainties ought to change under the constraint ofcertain diachronic axioms, now known as theAGM axioms in thebelief revision literature.^[9] For some reasons against the Harper-Levi approach to norms ofcertainty change, see Titelbaum (2013a: sec. 7.4.1).

5.5 Self-Locating Credences

One’sself-locating credences are, for example,credences about who one is, where one is, and what time it is. Suchcredences pose some challenges to conditionalization. Let me mentiontwo below.

To begin with, consider the following case, adapted from Titelbaum(2013a: ch. 12):

Example (Writer). At $t_1$ it’s middayon Wednesday, and a writer is sitting in an office finishing amanuscript for a publisher, with a deadline by the end of next day,being certain that she only has three more sections to go. Then, at$t_2$, she notices that it gets dark out—in fact, she has lostsense of time because of working too hard, and she is now only surethat it is either Wednesday evening or early Thursday morning. Shealso notices that she has only got one section done since the midday.So the writer utters to herself: “Now, I still have two moresections to go”. That is the new evidence for her to changecredences.

The problem is that it is not immediately clear what exactly is thepropositionE that the writer should conditionalize on. TherightE appears to be the proposition expressed by thewriter’s utterance: “Now, I still have two more sectionsto go”. And the expressed proposition must be one of thefollowing two candidates, depending on when the utterance is actuallymade (assuming the standard account of indexicals, due to Kaplan1989):

$(A)$: The writer still has two more sections to go on Wednesdayevening.

$(B)$: The writer still has two more sections to go on early ThursdayMorning.

But, with the lost sense of time, it also seems that the writer shouldconditionalize on a less informative body of evidence: the disjunction$A \vee B$. So exactly what should she conditionalize on? $A$,$B$, or $A \vee B$? See Titelbaum (2016) for a survey of someproposed solutions to this problem.

While the previous problem concerns only the inputs that should bepassed to the conditionalization process, conditionalization itself ischallenged when self-locating credences meet the danger of memoryloss. Consider the following case, made popular in epistemology byElga (2000):

Example (Sleeping Beauty). Sleeping Beautyparticipates in an experiment. She knows for sure that she will begiven a sleeping pill that induces limited amnesia. She knows for surethat, after she falls asleep, a fair coin will be flipped. If it landsheads, she will be awakened on Monday and asked: “How confidentare you that the coin landed heads?”. She will not be informedwhich day it is. If the coin lands tails, she will be awaken on bothMonday and on Tuesday and asked the same question each time. Theamnesia effect is designed to ensure that, if awakened on Tuesday shewill not remember being woken on Monday. And Sleeping Beauty knows allthat for sure.

What should her answer be when she is awakened on Monday and asked howconfident she is in the coin’s landing heads? Lewis (2001)employs the Principle of Conditionalization to argue that the answeris $1/2$. His reasoning proceeds as follows: Sleeping Beauty, uponher awakening, acquires no new evidence or acquires only a piece ofnew evidence that she is already certain of, so by conditionalizationher credence in the coin’s landing heads ought to remain thesame as it was before the sleep: $1/2$.

But Elga (2000) argues that the answer is $1/3$ rather than $1/2$.If so, that will seem to be a counterexample to the Principle ofConditionalization. Here is a sketch of his argument. Imagine that weare Sleeping Beauty and reason as follows. We just woke up, and thereare only three possibilities on the table, regarding how the coinlanded and what day it is today:

$(A)$: Heads and it’s Monday.

$(B)$: Tails and it’s Monday.

$(C)$: Tails and it’s Tuesday.

If we are told that it’s Monday ($A \vee B$), we will judgethat the coin’s landing heads ($A$) is as probable as itslanding tails ($B$). So

\[\Cr(A \mid A \vee B) = \Cr(B \mid A \vee B) = 1/2.\]

If we are told that it lands tails ($B \vee C$), we will judge thattoday being Monday ($B$) and today being Tuesday ($C$) are equallyprobable. So

\[\Cr(B \mid B \vee C) = \Cr(C \mid B \vee C) = 1/2.\]

The only way to meet the above conditions is to distribute theunconditional credences evenly:

\[\Cr(A) = \Cr(B) = \Cr(C) = 1/3.\]

Hence the credence in landing heads, $A$, is equal to $1/3$, or soElga concludes. This result seems to challenge the Principle ofConditionalization, which recommends the answer $1/2$ as explainedabove. For more on the Sleeping Beauty problem, see the survey byTitelbaum (2013b).

5.6 Bayesianism without Kinematics

Confronted with the existing problems for the Principle ofConditionalization, some Bayesians turn away from any diachronic normand develop another variety of Bayesianism:time-sliceBayesianism. On this view, what credences you should (or may)have at any particular timedepend solely on the totalevidence you have at that same time—independently of yourearlier credences. To specify this dependency relation is to specifyexclusively synchronic norms—and to forget about diachronicnorms. Strictly speaking, there is still a diachronic norm, but it isderived rather than fundamental: when the time flows from $t$ to$t'$, your credences ought to change in a certain way—theyought to change to the credences that you ought to have with respectto your total evidence at the latter time $t'$—and the earliertime $t$ is to be ignored. Any diachronic norm, if correct, is atmost an epiphenomenon that arises when correct synchronic norms areapplied repeatedly across different times, according to time-sliceBayesianism. (This view is stated above in terms of one’s totalevidence, but that can be replaced by one’s total reasons orinformation.)

A particular version of this view is held by J. Williamson (2010: ch.4), who is so firmly an objective Bayesian that he argues that thePrinciple of Conditionalization should be rejected if it is inconflict with repeated applications of certain synchronic norms, suchas Probabilism and the Principle of Maximum Entropy (which generalizesthe Principle of Indifference; seesupplement D). Time-slice Bayesianism as a general position is developed anddefended by Hedden (2015a, 2015b).

6. The Problem of Idealization

A worry about Bayesian epistemology is that the two core Bayesiannorms are so demanding that they can be followed only by highlyidealized agents—beinglogically omniscient, withprecise credences that always fit togetherperfectly. This is the problem of idealization, which waspresented in the tutorialsection 1.9. This section surveys three reply strategies for Bayesians, whichmight complement each other. As will become clear below, the work onthis problem is quite interdisciplinary, with contributions fromepistemologists as well as scientists and other philosophers.

6.1 De-idealization and Understanding

One reply to the problem of idealization is to look at how idealizedmodels are used and valued in science, and to argue that certainvalues of idealization can be carried over to epistemology. When ascientist studies a complex system, she might not really need anaccurate description of it but might rather want to pursue thefollowing:

some simplified, idealized models of the whole (such as a blocksliding on a frictionless, perfectly flat plane in vacuum);
gradual de-idealizations of the above (such as adding more andmore realistic considerations about friction);
an articulated reason why de-idealizations should be done this wayrather than another to improve upon the simpler models.

Parts 1 and 2 do not have to be ladders that will be kicked away oncewe reach a more realistic model. Instead, the three parts, 1–3,might work together to help the scientist achieve a deeperunderstanding of the complex system under study—a kind ofunderstanding that an accurate description (alone) does not provide.The above is one of the alleged values of idealized models inscientific modeling; for more, see section 4.2 of the entry onunderstanding and the survey by Elliott-Graves and Weisberg (2014: sec. 3). SomeBayesians have argued that certain values of idealization areapplicable not just in science but also in epistemology (Howson 2000:173–177; Titelbaum 2013a: ch. 2–5; Schupbach 2018). Formore on the values of building more or less idealized models not justin epistemology but generally in philosophy, see T. Williamson(2017).

The above reply to the problem of idealization has been reinforced bya sustained project of de-idealization in Bayesian epistemology. Thefollowing gives you the flavor of how this project may be pursued.Let’s start with the usual complaint that Probabilismimplies:

Strong Normalization. An agent ought to assigncredence 1 to every logical truth.

The worry is that a person can meet this demand only by luck or withan unrealistic ability—the ability to demarcate all logicaltruths from the other propositions. But some Bayesians argue that thestandard version of Probabilism can be suitably de-idealized to obtaina weak version that does not imply Strong Normalization. For example,the extensibility version of Probabilism (discussed insection 3.1) permits one to have credence gaps and, thus, have no credence in anylogical truth (de Finetti 1970 [1974]; Jeffrey 1983; Zynda 1996).Indeed, the extensibility version of Probabilism only implies:

Weak Normalization. It ought to be that, if anagent has a credence in a logical truth, that credence is equal to1.

Some Bayesians have tried to de-idealize Probabilism further, to setit free from the commitment that any credence ought to be as sharp asan individual real number, precise to every digit. For example, Walley(1991: ch. 2 and 3) develops a version of Probabilism according towhich a credence is permitted to be unsharp in this way. A credencecan be bounded by one or another interval of real numberswithout being equal to any particular real number or anyparticular interval—even the tightest bound on a credence can beanincomplete description of that credence. Thisinterval-bound approach gives rise to a Dutch Book argument for aneven weaker version of Probabilism, which only implies:

Very Weak Normalization. It ought to be that,if an agent has a credence in a logical truth, then that credence isbounded only by intervals that include 1.

Seesupplement A for some non-technical details. For more details and relatedcontroversies, see the survey by Mahtani (2019) and the entry onimprecise probabilities.

The above are just some of the possible steps that might be taken inthe Bayesian project of de-idealization. There are more: Can Bayesiansprovide norms for agents who can lose memories and forget what theyused to take as certain? See Meacham (2010), Moss (2012), andTitelbaum (2013a: ch. 6 and 7) for positive accounts; also seesection 5.4 for discussion. Can Bayesians develop norms for agents who aresomewhat incoherent and incapable of being perfectly coherent? SeeStaffel (2019) for a positive account. Can Bayesians provide normseven for agents who are so cognitively underpowered that they onlyhave all-or-nothing beliefs without a numerical credence? See Lin(2013) for a positive account. Can Bayesians develop norms thatexplain how one may be rationally uncertain whether one is rational?See Dorst (2020) for a positive account. Can Bayesians develop adiachronic norm for cognitively bounded agents? See Huttegger (2017a,2017b) for a positive account.

While the project of de-idealization can be pursued gradually andincrementally as illustrated above, Bayesians disagree about how farthis project should be pursued. Some Bayesians want to push itfurther: they think that Very Weak Normalization is still too strongto be plausible, so Probabilism needs to be abandoned altogether andreplaced by a norm that permits credences less than 1 in logicaltruths. For example, Garber (1983) tries to do that for certainlogical truths; Hacking (1967) and Talbott (2016), for all logicaltruths. On the other hand, Bayesians of the more traditional varietyretain a more or less de-idealized version of Probabilism, and try todefend it by clarifying its normative content, to which I nowturn.

6.2 Striving for Ideals

Probabilism is often thought to have a counterexample to this effect:it implies that we should meet a very high standard, but it is not thecase that we should, because we cannot. In reply, some Bayesians holdthat this is actually not a counterexample, and that the apparentcounterexample can be explained away once an appropriate reading of‘ought’ is in place and clearly distinguished from anotherreading.

To see that there are two readings of ‘ought’, think aboutthe following scenario. Suppose that this is true:

(i) We ought to launch a war now.

The truth of this particular norm might sound like a counterexample tothe general norm below:

(ii) There ought to be no war.

But perhaps there can be a context in which (i) and (ii) are both trueand hence the former is not a counterexample to the latter. An exampleis the context in which we know for sure that we are able to launch awar that ends all existing wars. Indeed, the occurrences of‘ought’ in those two sentences seem to have very differentreadings. Sentence (ii) can be understood to express a norm whichportrays what the state of the worldought to belike—what the world would be like if things wereideal.Such a norm is often called anought-to-be norm orevaluative norm, pointing to one or another ideal. On theother hand, sentence (i) can be understood as a norm which specifieswhat an agentought to do in a less-than-ideal situation thatshe turns out to be in—possibly with the goal to improve theexisting situation and bring it closer to the ideal specified by anought-to-be norm, or at least to prevent the situation from gettingworse. This kind of norm is often called anought-to-do norm,adeliberative norm, or aprescriptive norm. So,although the truth of (i) can sound like a counterexample to (ii), thetension between the two seems to disappear with appropriate readingsof ‘ought’.

Similarly, suppose that an ordinary human has some incoherentcredences, and that it is not the case that she ought to remove theincoherence right away because she has not detected the incoherence.The norm just stated can be thought of as an ought-to-do norm and,hence, need not be taken as a counterexample to Probabilism construedas an ought-to-be norm:

Probabilism (Ought-to-Be Version). Itought to be that one’s credences fit together in theprobabilistic way.

The ought-to-be reading of ‘ought’ has been employedimplicitly or explicitly to defend Bayesian norms—not just byBayesian philosophers (Zynda 1996; Christensen 2004: ch. 6; Titelbaum2013a: ch. 3 and 4; Wedgwood 2014; Eder forthcoming), but also byBayesian psychologists (Baron 2012). The distinction between theought-to-be and the ought-to-do oughts is most often defended in thebroader context of normative studies, such as in deontic logic(Castañeda 1970; Horty 2001: sec. 3.3 and 3.4) and inmetaethics (Broome 1999; Wedgwood 2006; Schroeder 2011).

The ought-to-be construal of Probabilism still leaves us aprescriptive issue: How should a person go about detecting and fixingthe incoherence of one’s credences, noting that it is absurd tostrive for coherence at all costs? This is an issue aboutought-to-do/prescriptive norms, addressed by a prescriptive researchprogram in an area of psychology calledjudgment and decisionmaking. For a survey of that area, see Baron (2004, 2012) andElqayam & Evans (2013). In fact, many psychologists even thinkthat, for better or worse, this prescriptive program has become the“new paradigm” in the psychology of reasoning; forreferences, see Elqayam & Over (2013).

The prescriptive issue mentioned above raises some other questions.There is anempirical, computational question: What is theextent to which a human brain can approximate the Bayesian ideal ofsynchronic and diachronic coherence? See Griffiths, Kemp, &Tenenbaum (2008) for a survey of some recent results. And there arephilosophical questions: Why is it epistemically better for ahuman’s credences to be less incoherent? Speaking of beingless incoherent, how can we develop a measure of degrees ofincoherence? See de Bona & Staffel (2018) and Staffel (2019) forproposals.

6.3 Applications Empowered by Idealization

There is a third approach to the problem of idealization: to someBayesians, some aspects of the Bayesian idealization are to beutilized rather than removed, because it is those aspects ofidealization thatempower certain important applications ofBayesian epistemology in science. Here is the idea. Consider a humanscientist confronted with an empirical problem. When some hypotheseshave been stated for consideration and some data have been collected,there remains an inferential task—the task of inferring from thedata to one of the hypotheses. This inferential task can be done byhuman scientists alone, but it has been done increasingly often thisway: by developing a computer program (in Bayesian statistics) tosimulate an idealized Bayesian agent as if that agent were hired toperform the inferential task. The purpose of this inferential taskwould be undermined if what is simulated by the computer were acognitively underpowered agent who mimics the limited capacities ofhuman agents. Howson (1992: sec. 6) suggests that this inferentialtask is what Bayesian epistemology and Bayesian statistics were mainlydesigned for at the early stages of their development. See Fienberg(2006) for the historical development of Bayesian statistics.

So, on the above view, idealization is essential to the existingapplications of Bayesian epistemology in science. If so, the realissue is whether the kind of scientific inquiryempowered byBayesian idealization serves the purpose of the inferential taskbetter than do the non-Bayesian rivals, such as so-calledfrequentism andlikelihoodism in statistics. For acritical comparison of those three schools of thought aboutstatistical inference, see Sober (2008: ch. 1), Hacking (2016), andthe entry onphilosophy of statistics. For an introduction to both Bayesian statistics and frequentiststatistics written for philosophers, see Howson & Urbach (2006:ch. 5–8).

7. Closing: The Expanding Territory of Bayesianism

Bayesian epistemology, despite the problems presented above, has beenexpanding its scope of application. In addition to the more standard,older areas of application listed insection 1.3, the newer ones can be found in the entry onepistemic self-doubt, sections 5.1 and 5.4 of the entry ondisagreement, Adler (2006 [2017]: sec. 6.3), and sections 3.6 and 4 of the entry onsocial epistemology.

In their more recent works, Bayesians have also started to contributeto some epistemological issues that have traditionally been among themost central concerns for many non-Bayesians, especially for thoseimmersed in the epistemology of all-or-nothing beliefs. I wish toclose by giving four groups of examples.

Skeptical Challenges: Central to traditionalepistemology is the issue of how to address certain skepticalchallenges. The Cartesian skeptic thinks that we are not justified inbelieving that we are not a brain in a vat. Huemer (2016) and Shogenji(2018) have each developed a Bayesian argument against this variety ofskepticism. There is also the Pyrrhonian skeptic, who holds the viewthat no belief can be justified due to the regress problem ofjustification: once a belief is justified with a reason, that reasonis in need of justification, too, which kickstarts a regress. Anattempt to reply to this skeptic quickly leads to a difficult choiceamong three positions: first, foundationalism (roughly, that theregress can be stopped); second, coherentism (roughly, that it ispermissible for the regress of justifications to be circular); andthird, infinitism (roughly, that it is permissible for the regress ofjustifications to extendad infinitum). To that issueBayesians have made some contributions. For example, White (2006)develops a Bayesian argument against an influential version offoundationalism, followed by a reply from Weatherson (2007); for more,seesection 3.2 of the entry on formal epistemology. Klein & Warfield (1994) develop a probabilistic argument againstcoherentism, which initiates a debate joined by many Bayesians; formore, seesection 7 of the entry on coherentist theories of epistemic justification. Peijnenburg (2007) defends infinitism by developing a Bayesianversion of it. For more on the Cartesian and Pyrrhonian skepticalviews, see the entry onskepticism.
Theories of Knowledge and Justified Beliefs:While traditional epistemologists praise knowledge and haveextensively studied what turns a belief into knowledge, Moss (2013,2018) develops a Bayesian counterpart: she argues that a credence canalso be knowledge-like, a property that can be studied by Bayesians.Traditional epistemology also features a number of competing accountsof justified belief, and the possibilities of their Bayesiancounterparts have been explored by Dunn (2015) and Tang (2016). Formore on the prospects of such Bayesian counterparts, see Hájekand Lin (2017).
The Scientific Realism/Anti-Realism Debate:One of the most classic debates in philosophy of science is thatbetween scientific realism and anti-realism. The scientific realistcontends that science pursues theories are true literally or at leastapproximately, while the anti-realist denies that. An earlycontribution to this debate is van Fraassen’s (1989: part II)Bayesian argument against inference to the best explanation (IBE),which is often used by scientific realists to defend their view. SomeBayesians have joined the debate and try to save IBE instead; seesections 3.1 and 4 of the entry onabduction. Another influential defense of scientific realism proceeds with theso-calledno-miracle argument. (This argument runs roughly asfollows: scientific realism is correct because it is the onlyphilosophical view that does not render the success of science amiracle.) Howson (2000: ch. 3) and Magnus & Callender (2004)maintain that the no-miracle argument commits a fallacy that can bemade salient from a Bayesian perspective. In reply, Sprenger &Hartmann (2019: ch. 5) contend that Bayesian epistemology makespossible a better version of the no-miracle argument for scientificrealism. An anti-realist view is instrumentalism, which says thatscience only need to pursue theories that are useful for makingobservable predictions. Vassend (forthcoming) argues thatconditionalization can be generalized in a way that caters to both thescientific realist and the instrumentalist—regardless of whetherevidence should be utilized in science to help us pursue truth orusefulness.
Frequentist Concerns: Frequentists aboutstatistical inference design inference procedures for the purposes of,say, testing a working hypothesis, identifying the truth among a setof competing hypotheses, or producing accurate estimates of certainquantities. And they want to design procedures that inferreliably—with a low objective, physical chance ofmaking errors. Those concerns have been incorporated into Bayesianstatistics, leading to the Bayesian counterparts of some frequentistaccounts. In fact, those results have already appeared in standardtextbooks on Bayesian statistics, such as the influential one byGelman et al. (2014: sec. 4.4 and ch. 6). The line between frequentistand Bayesian statistics is blurring.

So, as can be seen from the many examples in I–IV, Bayesianshave been assimilating ideas and concerns from the epistemologicaltradition of all-or-nothing beliefs. In fact, there have also beenattempts to develop a joint epistemology—an epistemology foragents who have both credences and all-or-nothing beliefs at the sametime; for details, seesection 4.2 of the entry on formal representations of belief.

It is debatable which, if any, of the above topics can be adequatelyaddressed in Bayesian epistemology. But Bayesians have been expandingtheir territory and their momentum will surely continue.

Bibliography

Adler, Jonathan, 2006 [2017], “Epistemological Problems ofTestimony”,The Stanford Encyclopedia of Philosophy(Winter 2017 Edition), Edward N. Zalta (ed.), first written 2006. URL= <https://plato.stanford.edu/archives/win2017/entries/testimony-episprob/>.
Armendt, Brad, 1980, “Is There a Dutch Book Argument forProbability Kinematics?”,Philosophy of Science, 47(4):583–588. doi:10.1086/288958
Arntzenius, Frank, 2003, “Some Problems forConditionalization and Reflection”,Journal ofPhilosophy, 100(7): 356–370.doi:10.5840/jphil2003100729
Bacchus, Fahiem, Henry E. Kyburg Jr, and Mariam Thalos, 1990,“Against Conditionalization”,Synthese, 85(3):475–506. doi:10.1007/BF00484837
Baron, Jonathan, 2004, “Normative Models of Judgment andDecision Making”, inBlackwell Handbook of Judgment andDecision Making, Derek J. Koehler and Nigel Harvey (eds.),London: Blackwell, 19–36.
–––, 2012, “The Point of Normative Modelsin Judgment and Decision Making”,Frontiers inPsychology, 3: art. 577. doi:10.3389/fpsyg.2012.00577
Bartha, Paul, 2004, “Countable Additivity and the de FinettiLottery”,The British Journal for the Philosophy ofScience, 55(2): 301–321. doi:10.1093/bjps/55.2.301
Bayes, Thomas, 1763, “An Essay Towards Solving a Problem inthe Doctrine of Chances”,Philosophical Transactions of theRoyal Society of London, 53: 370–418. Reprinted 1958,Biometrika, 45(3–4): 296–315, with G. A.Barnard’s “Thomas Bayes: A Biographical Note”,Biometrika, 45(3–4): 293–295.doi:10.1098/rstl.1763.0053 doi:10.1093/biomet/45.3-4.296doi:10.1093/biomet/45.3-4.293 (note)
Belot, Gordon, 2013, “Bayesian Orgulity”,Philosophy of Science, 80(4): 483–503.doi:10.1086/673249
Berger, James, 2006, “The Case for Objective BayesianAnalysis”,Bayesian Analysis, 1(3): 385–402.doi:10.1214/06-BA115
Blackwell, David and Lester Dubins, 1962, “Merging ofOpinions with Increasing Information”,The Annals ofMathematical Statistics, 33(3): 882–886.doi:10.1214/aoms/1177704456
Bovens, Luc and Stephan Hartmann, 2004,BayesianEpistemology, Oxford: Oxford University Press.doi:10.1093/0199269750.001.0001
Briggs, R.A., 2019, “Conditionals”, in Pettigrew andWeisberg 2019: 543–590.
Briggs, R.A. and Richard Pettigrew, 2020, “AnAccuracy-Dominance Argument for Conditionalization”,Noûs, 54(1): 162–181. doi:10.1111/nous.12258
Broome, John, 1999, “Normative Requirements”,Ratio, 12(4): 398–419. doi:10.1111/1467-9329.00101
Carnap, Rudolf, 1945, “On Inductive Logic”,Philosophy of Science, 12(2): 72–97.doi:10.1086/286851
–––, 1955, “Statistical and InductiveProbability and Inductive Logic and Science” (leaflet),Brooklyn, NY: Galois Institute of Mathematics and Art.
–––, 1963, “Replies and SystematicExpositions”, inThe Philosophy of Rudolf Carnap, PaulArthur Schilpp (ed.), La Salle, IL: Open Court, 859–1013.
Castañeda, Hector-Neri, 1970, “On the Semantics ofthe Ought-to-Do”,Synthese, 21(3–4):449–468. doi:10.1007/BF00484811
Christensen, David, 1996, “Dutch-Book ArgumentsDepragmatized: Epistemic Consistency For Partial Believers”,Journal of Philosophy, 93(9): 450–479.doi:10.2307/2940893
–––, 2004,Putting Logic in Its Place:Formal Constraints on Rational Belief, Oxford: Oxford UniversityPress. doi:10.1093/0199263256.001.0001
de Bona, Glauber and Julia Staffel, 2018, “Why Be(Approximately) Coherent?”,Analysis, 78(3):405–415. doi:10.1093/analys/anx159
de Finetti, Bruno, 1937,“La Prévision: Ses LoisLogiques, Ses Sources Subjectives”,Annales del’institut Henri Poincaré, 7(1):1–68.Translated as “Foresight: its Logical Laws, its SubjectiveSources”, Henry E. .Kyburg, Jr. (trans.), inStudies inSubjective Probability, Henry Ely Kyburg and Henry Edward Smokler(eds), New York: Wiley, 1964, 97–158. Second edition,Huntington: Robert Krieger, 1980, 53–118.
–––, 1970 [1974],Teoria delleprobabilità, Torino: G. Einaudi. Translated asTheoryof Probability, two volumes, Antonio Machi and Adrian Smith(trans), New York: John Wiley, 1974.
Diaconis, Persi and David Freedman, 1986a, “On theConsistency of Bayes Estimates”,The Annals ofStatistics, 14(1): 1–26. doi:10.1214/aos/1176349830
–––, 1986b, “Rejoinder: On the Consistencyof Bayes Estimates”,The Annals of Statistics, 14(1):63–67. doi:10.1214/aos/1176349842
Dorling, Jon, 1979, “Bayesian Personalism, the Methodologyof Scientific Research Programmes, and Duhem’s Problem”,Studies in History and Philosophy of Science Part A, 10(3):177–187. doi:10.1016/0039-3681(79)90006-2
Dorst, Kevin, 2020, “Evidence: A Guide for theUncertain”,Philosophy and Phenomenological Research,100(3): 586–632. doi:10.1111/phpr.12561
Duhem, Pierre, 1906 [1954],La théorie physique: sonobjet et sa structure, Paris: Chevalier & Rivière.Translated asThe Aim and Structure of Physical Theory,Philip P. Wiener (trans.), Princeton, NJ: Princeton University Press,1954.
Dunn, Jeff, 2015, “Reliability for Degrees of Belief”,Philosophical Studies, 172(7): 1929–1952.doi:10.1007/s11098-014-0380-2
Earman, John (ed.), 1983,Testing Scientific Theories,(Minnesota Studies in the Philosophy of Science 10), Minneapolis, MN:University of Minnesota Press.
–––, 1992,Bayes or Bust? A CriticalExamination of Bayesian Confirmation Theory, Cambridge, MA: MITPress.
Easwaran, Kenny, 2011, “Bayesianism I: Introduction andArguments in Favor”,Philosophy Compass, 6(5):312–320. doi:10.1111/j.1747-9991.2011.00399.x
–––, 2013, “Why CountableAdditivity?”,Thought: A Journal of Philosophy, 2(1):53–61. doi:10.1002/tht3.60
–––, 2014, “Regularity and HyperrealCredences”,Philosophical Review, 123(1): 1–41.doi:10.1215/00318108-2366479
–––, 2019, “ConditionalProbabilities”, in Pettigrew and Weisberg 2019:131–198.
Eder, Anna-Maria, forthcoming, “Evidential Probabilities andCredences”,The British Journal for the Philosophy ofScience, first online: 24 December 2020.doi:10.1093/bjps/axz043
Elga, Adam, 2000, “Self-Locating Belief and the SleepingBeauty Problem”,Analysis, 60(2): 143–147.doi:10.1093/analys/60.2.143
Elliott-Graves, Alkistis and Michael Weisberg, 2014,“Idealization”,Philosophy Compass, 9(3):176–185. doi:10.1111/phc3.12109
Elqayam, Shira and Jonathan St. B. T. Evans, 2013,“Rationality in the New Paradigm: Strict versus Soft BayesianApproaches”,Thinking & Reasoning, 19(3–4):453–470. doi:10.1080/13546783.2013.834268
Elqayam, Shira and David E. Over, 2013, “New ParadigmPsychology of Reasoning: An Introduction to the Special Issue Editedby Elqayam, Bonnefon, and Over”,Thinking &Reasoning, 19(3–4): 249–265.doi:10.1080/13546783.2013.841591
Eriksson, Lina and Alan Hájek, 2007, “What AreDegrees of Belief?”,Studia Logica, 86(2):183–213. doi:10.1007/s11225-007-9059-4
Eva, Benjamin, 2019, “Principles of Indifference”,The Journal of Philosophy, 116(7): 390–411.doi:10.5840/jphil2019116724
Eva, Benjamin and Stephan Hartmann, 2020, “On the Origins ofOld Evidence”,Australasian Journal of Philosophy,98(3): 481–494. doi:10.1080/00048402.2019.1658210
Fienberg, Stephen E., 2006, “When Did Bayesian InferenceBecome ‘Bayesian’?”,Bayesian Analysis,1(1): 1–40. doi:10.1214/06-BA101
Fishburn, Peter C., 1986, “The Axioms of SubjectiveProbability”,Statistical Science, 1(3): 335–345.doi:10.1214/ss/1177013611
Fitelson, Branden, 2006, “Inductive Logic”, inThePhilosophy of Science: An Encyclopedia, Sahotra Sarkar andJessica Pfeifer (eds), New York: Routledge, 384–394.
Fitelson, Branden and Andrew Waterman, 2005, “BayesianConfirmation and Auxiliary Hypotheses Revisited: A Reply toStrevens”,The British Journal for the Philosophy ofScience, 56(2): 293–302. doi:10.1093/bjps/axi117
Foley, Richard, 1992,Working without a Net: A Study ofEgocentric Epistemology, New York: Oxford University Press.
Forster, Malcolm R., 1995, “Bayes and Bust: Simplicity as aProblem for a Probabilist’s Approach to Confirmation”,The British Journal for the Philosophy of Science, 46(3):399–424. doi:10.1093/bjps/46.3.399
Forster, Malcolm and Elliott Sober, 1994, “How to Tell WhenSimpler, More Unified, or LessAd Hoc Theories Will ProvideMore Accurate Predictions”,The British Journal for thePhilosophy of Science, 45(1): 1–35.doi:10.1093/bjps/45.1.1
Freedman, David A., 1963, “On the Asymptotic Behavior ofBayes’ Estimates in the Discrete Case”,The Annals ofMathematical Statistics, 34(4): 1386–1403.doi:10.1214/aoms/1177703871
Gabbay, Dov M., Stephan Hartman, and John Woods (eds), 2011,Handbook of the History of Logic, Volume 10: Inductive Logic,Boston: Elsevier.
Gaifman, Haim, 1986, “ A Theory of Higher OrderProbabilities”,Proceedings of the 1986 Conference onTheoretical Aspects of Reasoning about Knowledge, San Francisco:Morgan Kaufmann Publishers, 275–292.
Gaifman, Haim and Marc Snir, 1982, “Probabilities over RichLanguages, Testing and Randomness”,Journal of SymbolicLogic, 47(3): 495–548. doi:10.2307/2273587
Garber, Daniel, 1983, “Old Evidence and Logical Omnisciencein Bayesian Confirmation Theory”, in Earman 1983: 99–131. [Garber 1983 available online]
Gelman, Andrew, John B. Carlin, Hal Steven Stern, David B. Dunson,Aki Vehtari, and Donald B. Rubin, 2014,Bayesian DataAnalysis, third edition, (Chapman & Hall/CRC Texts inStatistical Science), Boca Raton, FL: CRC Press.
Gelman, Andrew and Christian Hennig, 2017, “BeyondSubjective and Objective in Statistics”,Journal of theRoyal Statistical Society: Series A (Statistics in Society),180(4): 967–1033. Includes discussions of the paper.doi:10.1111/rssa.12276
Gendler, Tamar Szabo and John Hawthorne (eds), 2010,OxfordStudies in Epistemology, Volume 3, Oxford: Oxford UniversityPress.
Gillies, Donald, 2000,Philosophical Theories ofProbability, (Philosophical Issues in Science), London/New York:Routledge.
Glymour, Clark N., 1980, “Why I Am Not a Bayesian”, inhisTheory and Evidence, Princeton, NJ: Princeton UniversityPress.
Good, Irving John, 1976, “The Bayesian Influence, or How toSweep Subjectivism under the Carpet”, inFoundations ofProbability Theory, Statistical Inference, and Statistical Theories ofScience, William Leonard Harper and Clifford Alan Hooker (eds.),Dordrecht: Springer Netherlands, 125–174. Reprinted in hisGood Thinking: The Foundations of Probability and ItsApplications, Minneapolis, MN: University of Minnesota Press,22–58. doi:10.1007/978-94-010-1436-6_5
Goodman, Nelson, 1955,Fact, Fiction, and Forecast,Cambridge, MA: Harvard University Press.
Greaves, Hilary and David Wallace, 2006, “JustifyingConditionalization: Conditionalization Maximizes Expected EpistemicUtility”,Mind, 115(459): 607–632.doi:10.1093/mind/fzl607
Griffiths, Thomas L., Charles Kemp, and Joshua B. Tenenbaum, 2008,“Bayesian Models of Cognition”, inThe CambridgeHandbook of Computational Psychology, Ron Sun (ed.), Cambridge:Cambridge University Press, 59–100.doi:10.1017/CBO9780511816772.006
Hacking, Ian, 1967, “Slightly More Realistic PersonalProbability”,Philosophy of Science, 34(4):311–325. doi:10.1086/288169
–––, 2001,An Introduction to Probabilityand Inductive Logic, Cambridge: Cambridge University Press.doi:10.1017/CBO9780511801297
–––, 2016,Logic of StatisticalInference, Cambridge: Cambridge University Press.doi:10.1017/CBO9781316534960
Hájek, Alan, 2003, “What Conditional ProbabilityCould Not Be”,Synthese, 137(3): 273–323.doi:10.1023/B:SYNT.0000004904.91112.16
–––, 2009, “Dutch Book Arguments”,inThe Handbook of Rational and Social Choice, Paul Anand,Prasanta Pattanaik, and Clemens Puppe (eds.), New York: OxfordUniversity Press, 173–195.doi:10.1093/acprof:oso/9780199290420.003.0008
–––, 2012, “Is Strict CoherenceCoherent?”,Dialectica, 66(3): 411–424.doi:10.1111/j.1746-8361.2012.01310.x
Hájek, Alan and Hanti Lin, 2017, “A Tale of TwoEpistemologies?”,Res Philosophica, 94(2):207–232.
Harman, Gilbert, 1986,Change in View: Principles ofReasoning, Cambridge, MA: MIT Press.
Harsanyi, John C., 1985, “Acceptance of EmpiricalStatements: A Bayesian Theory without Cognitive Utilities”,Theory and Decision, 18(1): 1–30.
Harper, William L., 1976, “Rational ConceptualChange”,PSA: Proceedings of the Biennial Meeting of thePhilosophy of Science Association, 1976(2): 462–494.doi:10.1086/psaprocbienmeetp.1976.2.192397
–––, 1978, “Bayesian Learning Models withRevision of Evidence”,Philosophia, 7(2):357–367. doi:10.1007/BF02378821
Hartmann, Stephan and Branden Fitelson, 2015, “A NewGarber-Style Solution to the Problem of Old Evidence”,Philosophy of Science, 82(4): 712–717.doi:10.1086/682916
Haverkamp, Nick and Moritz Schulz, 2012, “A Note onComparative Probability”,Erkenntnis, 76(3):395–402. doi:10.1007/s10670-011-9307-x
Heckerman, David, 1996 [2008], “A Tutorial on Learning withBayesian Networks”. Technical Report MSR-TR-95-06, Redmond, WA:Microsoft Research. Reprinted inInnovations in Bayesian Networks:Theory and Applications, Dawn E. Holmes and Lakhmi C. Jain(eds.), (Studies in Computational Intelligence, 156),Berlin/Heidelberg: Springer Berlin Heidelberg, 2008, 33–82.doi:10.1007/978-3-540-85066-3_3
Hedden, Brian, 2015a, “Time-Slice Rationality”,Mind, 124(494): 449–491. doi:10.1093/mind/fzu181
–––, 2015b,Reasons without Persons:Rationality, Identity, and Time, Oxford/New York: OxfordUniversity Press. doi:10.1093/acprof:oso/9780198732594.001.0001
Henderson, Leah, 2014, “Bayesianism and Inference to theBest Explanation”,The British Journal for the Philosophy ofScience, 65(4): 687–715. doi:10.1093/bjps/axt020
Hitchcock, Christopher (ed.), 2004,Contemporary Debates inPhilosophy of Science, (Contemporary Debates in Philosophy 2),Malden, MA: Blackwell.
Horgan, Terry, 2017, “Troubles for Bayesian FormalEpistemology”,Res Philosophica, 94(2): 233–255.doi:10.11612/resphil.1535
Horty, John F., 2001,Agency and Deontic Logic,Oxford/New York: Oxford University Press.doi:10.1093/0195134613.001.0001
Howson, Colin, 1992, “Dutch Book Arguments andConsistency”,PSA: Proceedings of the Biennial Meeting ofthe Philosophy of Science Association, 1992(2): 161–168.doi:10.1086/psaprocbienmeetp.1992.2.192832
–––, 2000,Hume’s Problem: Inductionand the Justification of Belief, Oxford: Clarendon Press.
Howson, Colin and Peter Urbach, 2006,Scientific Reasoning:The Bayesian Approach, third edition, Chicago: Open Court. Firstedition, 1989.
Huber, Franz, 2018,A Logical Introduction to Probability andInduction, New York: Oxford University Press.
Huemer, Michael, 2016, “Serious Theories and SkepticalTheories: Why You Are Probably Not a Brain in a Vat”,Philosophical Studies, 173(4): 1031–1052.doi:10.1007/s11098-015-0539-5
Hume, David, 1748/1777 [2008],An Enquiry Concerning HumanUnderstanding, London. Last edition corrected by the author,1777. 1777 edition reprinted, Peter Millican (ed.), (OxfordWorld’s Classics), New York/Oxford: Oxford University Press.
Huttegger, Simon M., 2015, “Merging of Opinions andProbability Kinematics”,The Review of Symbolic Logic,8(4): 611–648. doi:10.1017/S1755020315000180
–––, 2017a, “Inductive Learning in Smalland Large Worlds”,Philosophy and PhenomenologicalResearch, 95(1): 90–116. doi:10.1111/phpr.12232
–––, 2017b,The Probabilistic Foundations ofRational Learning, Cambridge: Cambridge University Press.doi:10.1017/9781316335789
Ivanova, Milena, 2021,Duhem and Holism, Cambridge:Cambridge University Press. doi:10.1017/9781009004657
Jaynes, Edwin T., 1957, “Information Theory and StatisticalMechanics”,Physical Review, 106(4): 620–630.doi:10.1103/PhysRev.106.620
–––, 1968, “Prior Probabilities”,IEEE Transactions on Systems Science and Cybernetics, 4(3):227–241. doi:10.1109/TSSC.1968.300117
–––, 1973, “The Well-Posed Problem”,Foundations of Physics, 3(4): 477–492.doi:10.1007/BF00709116
Jeffrey, Richard C., 1965 [1983],The Logic of Decision,(McGraw-Hill Series in Probability and Statistics), New York:McGraw-Hill. Second edition, Chicago: University of Chicago Press,1983.
–––, 1970, “Dracula Meets Wolfman:Acceptance vs. Partial Belief”, inInduction, Acceptance andRational Belief, Marshall Swain (ed.), Dordrecht: SpringerNetherlands, 157–185. doi:10.1007/978-94-010-3390-9_8
–––, 1983, “Bayesianism with a HumanFace”, in Earman 1983: 133–156. [Jeffrey 1983 available online]
–––, 1986, “Probabilism andInduction”,Topoi, 5(1): 51–58.doi:10.1007/BF00137829
Jeffreys, Harold, 1939,Theory of Probability, Oxford:Oxford University Press.
–––, 1946, “An Invariant Form for thePrior Probability in Estimation Problems”,Proceedings ofthe Royal Society of London. Series A. Mathematical and PhysicalSciences, 186(1007): 453–461.doi:10.1098/rspa.1946.0056
Joyce, James M., 1998, “A Nonpragmatic Vindication ofProbabilism”,Philosophy of Science, 65(4):575–603. doi:10.1086/392661
–––, 1999,The Foundations of CausalDecision Theory, Cambridge: Cambridge University Press.doi:10.1017/CBO9780511498497
–––, 2003 [2021], “Bayes’Theorem”,The Stanford Encyclopedia of Philosophy (Fall2021 edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2021/entries/bayes-theorem/>
–––, 2005, “How Probabilities ReflectEvidence”,Philosophical Perspectives, 19(1):153–178. doi:10.1111/j.1520-8583.2005.00058.x
–––, 2011, “The Development of SubjectiveBayesianism”, in Gabbay, Hartmann, and Woods 2011:415–475. doi:10.1016/B978-0-444-52936-7.50012-4
Kaplan, David, 1989, “Demonstratives. An Essay on theSemantics, Logic, Metaphysics, and Epistemology of Demonstratives andOther Indexicals”, inThemes from Kaplan, Joseph Almog,John Perry, and Howard Wettstein (eds.), New York: Oxford UniversityPress, 481–563.
Kass, Robert E. and Larry Wasserman, 1996, “The Selection ofPrior Distributions by Formal Rules”,Journal of theAmerican Statistical Association, 91(435): 1343–1370.
Kelly, Kevin T., 1996,The Logic of Reliable Inquiry,(Logic and Computation in Philosophy), New York: Oxford UniversityPress.
–––, 2000, “The Logic of Success”,The British Journal for the Philosophy of Science, 51(S1):639–666. doi:10.1093/bjps/51.4.639
Kelly, Kevin T., and Clark Glymour, 2004, “Why ProbabilityDoes Not Capture the Logic of Scientific Justification”, inHitchcock 2004: 94–114.
Kemeny, John G., 1955, “Fair Bets and InductiveProbabilities”,Journal of Symbolic Logic, 20(3):263–273. doi:10.2307/2268222
Keynes, John Maynard, 1921,A Treatise on Probability,London: Macmillan.
Klein, Peter and Ted A. Warfield, 1994, “What PriceCoherence?”,Analysis, 54(3): 129–132.doi:10.1093/analys/54.3.129
Kolmogorov, A. N., 1933,Grundbegriffe derWahrscheinlichkeitsrechnung, Berlin: Springer. Translated asFoundations of the Theory of Probability, Nathan Morrison(ed.), New York: Chelsea, 1950. Second English edition with an addedbibliography by A.T. Bharucha-Reid, New York: Chelsea, 1956. Secondedition reprinted Mineola, NY: Dover, 2018.
Konek, Jason, 2019, “Comparative Probabilities”, inPettigrew and Weisberg 2019: 267–348.
–––, forthcoming, “The Art ofLearning”, inOxford Studies in Epistemology, Volume 7,Oxford: Oxford University Press.
Kopec, Matthew and Michael G. Titelbaum, 2016, “TheUniqueness Thesis”,Philosophy Compass, 11(4):189–200. doi:10.1111/phc3.12318
Laplace, Pierre Simon, 1814 [1902],Essai philosophique surles probabilités, Paris: Mme. Ve. Courcier. Translated asA Philosophical Essay on Probabilities, Frederick WilsonTruscott and Frederick Lincoln Emory (trans.), New York: J. Wiley,1902.
Levi, Isaac, 1980,The Enterprise of Knowledge: An Essay onKnowledge, Credal Probability, and Chance, Cambridge, MA: MITPress.
Lewis, David, 1980, “A Subjectivist’s Guide toObjective Chance”, inStudies in Inductive Logic andProbability, Volume 2, R.C. Jeffrey (ed.), Berkeley, CA:University of California Press, 263–293. Reprinted inLewis’sPhilosophical Papers, Volume 2, Oxford: OxfordUniversity Press, 1986, ch. 19.
–––, 1999, “Why Conditionalize?”, inhisPapers in Metaphysics and Epistemology, Cambridge:Cambridge University Press, 403–407.
–––, 2001, “Sleeping Beauty: Reply toElga”,Analysis, 61(3): 171–176.doi:10.1093/analys/61.3.171
Lin, Hanti, 2013, “Foundations of Everyday PracticalReasoning”,Journal of Philosophical Logic, 42(6):831–862. doi:10.1007/s10992-013-9296-0
–––, forthcoming, “Modes of Convergence tothe Truth: Steps toward a Better Epistemology of Induction”,The Review of Symbolic Logic, first online: 3 January 2022.doi:10.1017/S1755020321000605
Lipton, Peter, 2004,Inference to the Best Explanation,second edition, (International Library of Philosophy), London/NewYork: Routledge/Taylor and Francis Group.
Magnus, P. D. and Craig Callender, 2004, “Realist Ennui andthe Base Rate Fallacy”,Philosophy of Science, 71(3):320–338. doi:10.1086/421536
Maher, Patrick, 1992, “Diachronic Rationality”,Philosophy of Science, 59(1): 120–141.doi:10.1086/289657
–––, 2004, “Probability Captures the Logicof Scientific Confirmation”, in Hitchcock 2004:69–93.
Mahtani, Anna, 2019, “Imprecise Probabilities”, inPettigrew and Weisberg 2019: 107–130.
Meacham, Chris J.G., 2010, “Unravelling the Tangled Web:Continuity, Internalism, Non-uniqueness and Self-LocatingBeliefs”, in Gendler and Hawthorne 2010: 86–125.
–––, 2015, “UnderstandingConditionalization”,Canadian Journal of Philosophy,45(5–6): 767–797. doi:10.1080/00455091.2015.1119611
–––, 2016, “Ur-Priors, Conditionalization,and Ur-Prior Conditionalization”,Ergo, an Open AccessJournal of Philosophy, 3: art. 17.doi:10.3998/ergo.12405314.0003.017
Morey, Richard D., Jan-Willem Romeijn, and Jeffrey N. Rouder,2013, “The Humble Bayesian: Model Checking from a Fully BayesianPerspective”,British Journal of Mathematical andStatistical Psychology, 66(1): 68–75.doi:10.1111/j.2044-8317.2012.02067.x
Moss, Sarah, 2012, “Updating as Communication”,Philosophy and Phenomenological Research, 85(2):225–248. doi:10.1111/j.1933-1592.2011.00572.x
–––, 2013, “EpistemologyFormalized”,Philosophical Review, 122(1): 1–43.doi:10.1215/00318108-1728705
–––, 2018,Probabilistic Knowledge,Oxford, United Kingdom: Oxford University Press.doi:10.1093/oso/9780198792154.001.0001
Niiniluoto, Ilkka, 1983, “Novel Facts andBayesianism”,The British Journal for the Philosophy ofScience, 34(4): 375–379. doi:10.1093/bjps/34.4.375
Okasha, Samir, 2000, “Van Fraassen’s Critique ofInference to the Best Explanation”,Studies in History andPhilosophy of Science Part A, 31(4): 691–710.doi:10.1016/S0039-3681(00)00016-9
Peijnenburg, Jeanne, 2007, “Infinitism Regained”,Mind, 116(463): 597–602. doi:10.1093/mind/fzm597
Peirce, Charles Sanders, 1877, “The Fixation ofBelief”,Popular Science Monthly, 12: 1–15.Reprinted in 1955,Philosophical Writings of Peirce, JustusBuchler (ed.), Dover Publications, 5–22.
–––, 1903, “The Three NormativeSciences”, fifth Harvard lecture on pragmatism delivered 30April 1903. Reprinted in 1998,The Essential Peirce, Vol. 2(1893–1913), The Peirce Edition Project (ed.), Bloomington,IN: Indiana University Press, 196–207 (ch. 14).
Pettigrew, Richard, 2012, “Accuracy, Chance, and thePrincipal Principle”,Philosophical Review, 121(2):241–275. doi:10.1215/00318108-1539098
–––, 2016,Accuracy and the Laws ofCredence, Oxford, United Kingdom: Oxford University Press.doi:10.1093/acprof:oso/9780198732716.001.0001
–––, 2020a,Dutch Book Arguments,Cambridge: Cambridge University Press. doi:10.1017/9781108581813
–––, 2020b, “What Is Conditionalization,and Why Should We Do It?”,Philosophical Studies,177(11): 3427–3463. doi:10.1007/s11098-019-01377-y
Pettigrew, Richard and Jonathan Weisberg (eds), 2019,The OpenHandbook of Formal Epistemology, PhilPapers Foundation. [Pettigrew and Weisberg (eds) 2019 available online]
Pollock, John L., 2006,Thinking about Acting: LogicalFoundations for Rational Decision Making, Oxford/New York: OxfordUniversity Press.
Popper, Karl R., 1959,The Logic of Scientific Discovery,New York: Basic Books. Reprinted, London: Routledge, 1992.
Putnam, Hilary, 1963, “Probability and Confirmation”,The Voice of America Forum Lectures, Philosophy of ScienceSeries, No. 10, Washington, D.C.: United States InformationAgency, pp. 1–11. Reprinted in hisMathematics, Matter, andMethod, London/New York: Cambridge University Press, 1975,293–304.
Quine, W. V., 1951, “Main Trends in Recent Philosophy: TwoDogmas of Empiricism”,The Philosophical Review, 60(1):20–43. doi:10.2307/2181906
Ramsey, Frank Plumpton, 1926 [1931], “Truth andProbability”, manuscript. Printed inFoundations ofMathematics and Other Logical Essays, R.B. Braithwaite (ed.),London: Kegan, Paul, Trench, Trubner & Co. Ltd., 1931,156–198.
Rawls, John, 1971,A Theory of Justice, Cambridge, MA:Harvard University Press. Revised edition 1999.
Reichenbach, Hans, 1938,Experience and Prediction: AnAnalysis of the Foundations and the Structure of Knowledge,Chicago: The University of Chicago Press.
Rényi, Alfréd, 1970,Foundations ofProbability, San Francisco: Holden-Day.
Rescorla, Michael, 2015, “Some Epistemological Ramificationsof the Borel–Kolmogorov Paradox”,Synthese, 192:735–767. doi:10.1007/s11229-014-0586-z
–––, 2018, “A Dutch Book Theorem andConverse Dutch Book Theorem for Kolmogorov Conditionalization”,The Review of Symbolic Logic, 11(4): 705–735.doi:10.1017/S1755020317000296
–––, 2021, “On the Proper Formulation ofConditionalization”,Synthese, 198(3): 1935–1965.doi:10.1007/s11229-019-02179-9
Rosenkrantz, Roger D., 1981,Foundations and Applications ofInductive Probability, Atascadero, CA: Ridgeview.
–––, 1983, “Why Glymour Is aBayesian”, in Earman 1983: 69–97. [Rosenkrantz 1983 available online]
Salmon, Wesley C., 1990, “Rationality and Objectivity inScience or Tom Kuhn Meets Tom Bayes”, inScientificTheories (Minnesota Studies in the Philosophy of Science, 14), C.W. Savage (ed.), Minneapolis, MN: University of Minnesota Press,175–205.
Savage, Leonard J., 1972,The Foundations of Statistics,second revised edtion, New York: Dover Publications.
Schoenfield, Miriam, 2014, “Permission to Believe: WhyPermissivism Is True and What It Tells Us About Irrelevant Influenceson Belief”,Noûs, 48(2): 193–218.doi:10.1111/nous.12006
Schroeder, Mark, 2011, “Ought, Agents, and Actions”,Philosophical Review, 120(1): 1–41.doi:10.1215/00318108-2010-017
Schupbach, Jonah N., 2018, “Troubles for Bayesian FormalEpistemology? A Response to Horgan”,Res Philosophica,95(1): 189–197. doi:10.11612/resphil.1652
Seidenfeld, Teddy, 1979, “Why I Am Not an ObjectiveBayesian; Some Reflections Prompted by Rosenkrantz”,Theoryand Decision, 11(4): 413–440. doi:10.1007/BF00139451
–––, 2001, “Remarks on the Theory ofConditional Probability: Some Issues of Finite Versus CountableAdditivity”, inProbability Theory: Philosophy, RecentHistory and Relations to Science, Vincent F. Hendricks, StigAndur Pedersen, and Klaus Frovin Jørgensen (eds.), (SyntheseLibrary 297), Dordrecht/Boston: Kluwer Academic Publishers,167–178.
Shimony, Abner, 1955, “Coherence and the Axioms ofConfirmation”,Journal of Symbolic Logic, 20(1):1–28. doi:10.2307/2268039
–––, 1970, “Scientific Inference”,inThe Nature and Function of Scientific Theories (PittsburghStudies in the Philosophy of Science, 4), Robert G. Colodny (ed.),Pittsburgh, PA: University of Pittsburgh Press, 79–172.
Shogenji, Tomoji, 2018,Formal Epistemology and CartesianSkepticism: In Defense of Belief in the Natural World, (RoutledgeStudies in Contemporary Philosophy 101), New York: Routledge, Taylor& Francis Group.
Skyrms, Brian, 1966 [2000],Choice and Chance: An Introductionto Inductive Logic, Belmont, CA: Dickenson. Fourth edition,Belmont, CA: Wadsworth, 2000.
–––, 1984,Pragmatics and Empiricism,New Haven, CT: Yale University Press.
Smith, Cedric A. B., 1961, “Consistency in StatisticalInference and Decision”,Journal of the Royal StatisticalSociety: Series B (Methodological), 23(1): 1–25.doi:10.1111/j.2517-6161.1961.tb00388.x
Sober, Elliott, 2002, “Bayesianism—Its Scope andLimits”, inBayes’s Theorem (Proceedings of theBritish Academy, 113), Richard Swinburne (ed.), Oxford: OxfordUniversity Press.
–––, 2008,Evidence and Evolution: The Logicbehind the Science, Cambridge: Cambridge University Press.doi:10.1017/CBO9780511806285
Sprenger, Jan, 2015, “A Novel Solution to the Problem of OldEvidence”,Philosophy of Science, 82(3): 383–401.doi:10.1086/681767
–––, 2018, “The Objectivity of SubjectiveBayesianism”,European Journal for Philosophy ofScience, 8(3): 539–558. doi:10.1007/s13194-018-0200-1
Sprenger, Jan and Stephan Hartmann, 2019,Bayesian Philosophyof Science: Variations on a Theme by the Reverend Thomas Bayes,Oxford/New York: Oxford University Press.doi:10.1093/oso/9780199672110.001.0001
Staffel, Julia, 2019,Unsettled Thoughts: A Theory of Degreesof Rationality, Oxford/New York: Oxford University Press.doi:10.1093/oso/9780198833710.001.0001
Stalnaker, Robert C., 1970, “Probability andConditionals”,Philosophy of Science, 37(1):64–80. doi:10.1086/288280
Stefánsson, H. Orri, 2017, “What Is‘Real’ in Probabilism?”,Australasian Journal ofPhilosophy, 95(3): 573–587.doi:10.1080/00048402.2016.1224906
Strevens, Michael, 2001, “The Bayesian Treatment ofAuxiliary Hypotheses”,The British Journal for thePhilosophy of Science, 52(3): 515–537.doi:10.1093/bjps/52.3.515
Talbott, William J., 1991, “Two Principles of BayesianEpistemology”,Philosophical Studies, 62(2):135–150. doi:10.1007/BF00419049
–––, 2016, “A Non-Probabilist Principle ofHigher-Order Reasoning”,Synthese, 193(10):3099–3145. doi:10.1007/s11229-015-0922-y
Tang, Weng Hong, 2016, “Reliability Theories of JustifiedCredence”,Mind, 125(497): 63–94.doi:10.1093/mind/fzv199
Teller, Paul, 1973, “Conditionalization andObservation”,Synthese, 26(2): 218–258.doi:10.1007/BF00873264
Titelbaum, Michael G., 2013a,Quitting Certainties: A BayesianFramework Modeling Degrees of Belief, Oxford: Oxford UniversityPress. doi:10.1093/acprof:oso/9780199658305.001.0001
–––, 2013b, “Ten Reasons to Care About theSleeping Beauty Problem”,Philosophy Compass, 8(11):1003–1017. doi:10.1111/phc3.12080
–––, 2016, “Self-LocatingCredences”, inThe Oxford Handbook of Probability andPhilosophy, Alan Hájek, and Christopher Hitchcock (eds),Oxford: Oxford University Press, p. 666–680.
–––, forthcoming,Fundamentals of BayesianEpistemology, Oxford University Press.
van Fraassen, Bas C., 1984, “Belief and the Will”,The Journal of Philosophy, 81(5): 235–256.doi:10.2307/2026388
–––, 1988, “The Problem of OldEvidence”, inPhilosophical Analysis, David F. Austin(ed.), Dordrecht: Springer Netherlands, 153–165.doi:10.1007/978-94-009-2909-8_10
–––, 1989,Laws and Symmetry,Oxford/New York: Oxford University Press.doi:10.1093/0198248601.001.0001
–––, 1995, “Belief and the Problem ofUlysses and the Sirens”,Philosophical Studies, 77(1):7–37. doi:10.1007/BF00996309
Vassend, Olav Benjamin, forthcoming, “Justifying the Normsof Inductive Inference”,The British Journal for thePhilosophy of Science, first online: 17 December 2020.doi:10.1093/bjps/axz041
von Mises, Richard, 1928 [1981],Wahrscheinlichkeit,Statistik, und Wahrheit, J. Springer; third German edition, 1951.Third edition translated asProbability, Statistics, andTruth, second revised edition, Hilda Geiringer (trans.), London:George Allen & Unwin, 1951. Reprinted New York: Dover, 1981.
Walley, Peter, 1991,Statistical Reasoning with ImpreciseProbabilities, London: Chapman and Hall.
Wasserman, Larry, 1998, “Asymptotic Properties ofNonparametric Bayesian Procedures”, inPracticalNonparametric and Semiparametric Bayesian Statistics, Dipak Dey,Peter Müller, and Debajyoti Sinha (eds.), (Lecture Notes inStatistics 133), New York: Springer New York, 293–304.doi:10.1007/978-1-4612-1732-9_16
Weatherson, Brian, 2007, “The Bayesian and theDogmatist”,Proceedings of the Aristotelian Society(Hardback), 107(1pt2): 169–185.doi:10.1111/j.1467-9264.2007.00217.x
Wedgwood, Ralph, 2006, “The Meaning of ‘Ought’”,Oxford Studies in Metaethics, Volume 1, RussShafer-Landau (ed.), Oxford: Clarendon Press, 127–160.
–––, 2014, “Rationality as a Virtue:Rationality as a Virtue”,Analytic Philosophy, 55(4):319–338. doi:10.1111/phib.12055
Weisberg, Jonathan, 2007, “Conditionalization, Reflection,and Self-Knowledge”,Philosophical Studies, 135(2):179–197. doi:10.1007/s11098-007-9073-4
–––, 2009a, “Locating IBE in the BayesianFramework”,Synthese, 167(1): 125–143.doi:10.1007/s11229-008-9305-y
–––, 2009b, “Commutativity or Holism? ADilemma for Conditionalizers”,The British Journal for thePhilosophy of Science, 60(4): 793–812.doi:10.1093/bjps/axp007
–––, 2011, “Varieties ofBayesianism”, in Gabbay, Hartmann, and Woods 2011:477–551. doi:10.1016/B978-0-444-52936-7.50013-6
Wenmackers, Sylvia, 2019, “InfinitesimalProbabilities”, in Pettigrew and Weisberg 2019:199–265.
Wenmackers, Sylvia and Jan-Willem Romeijn, 2016, “New Theoryabout Old Evidence: A Framework for Open-Minded Bayesianism”,Synthese, 193(4): 1225–1250.doi:10.1007/s11229-014-0632-x
White, Roger, 2006, “Problems for Dogmatism”,Philosophical Studies, 131(3): 525–557.doi:10.1007/s11098-004-7487-9
–––, 2010, “Evidential Symmetry and MushyCredence”, in Gendler and Hawthorne 2010: 161–186.
Williamson, Jon, 1999, “Countable Additivity and SubjectiveProbability”,The British Journal for the Philosophy ofScience, 50(3): 401–416. doi:10.1093/bjps/50.3.401
–––, 2010,In Defence of ObjectiveBayesianism, Oxford/New York: Oxford University Press.doi:10.1093/acprof:oso/9780199228003.001.0001
Williamson, Timothy, 2007, “How Probable Is an InfiniteSequence of Heads?”,Analysis, 67(3): 173–180.doi:10.1093/analys/67.3.173
–––, 2017, “Model-Building inPhilosophy”, inPhilosophy’s Future: The Problem ofPhilosophical Progress, Russell Blackford and Damien Broderick(eds.), Hoboken, NJ: Wiley, 159–171.doi:10.1002/9781119210115.ch12
Yalcin, Seth, 2012, “Bayesian Expressivism”,Proceedings of the Aristotelian Society (Hardback),112(2pt2): 123–160. doi:10.1111/j.1467-9264.2012.00329.x
Zynda, Lyle, 1996, “Coherence as an Ideal ofRationality”,Synthese, 109(2): 175–216.doi:10.1007/BF00413767

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Acknowledgments

I thank Alan Hájek for his incredibly extensive, extremelyhelpful comments. I thank G. J. Mattey for his long-term support andeditorial assistance. I also thank William Talbott, Stephan Hartmann,Jon Williamson, Chloé de Canson, Maomei Wang, Ted Shear, JeremyStrasser, Kramer Thompson, Joshua Thong, James Willoughby, RachelBoddy, and Tyrus Fisher for their comments and suggestions.

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Library of Congress Catalog Data: ISSN 1095-5054

	A is true	A is false
buy “win $100 ifA istrue” at $75	\(-\$75 + \$100\)	\(-\$75\)
buy “win $100 if \(\neg A\) istrue” at $30	\(-\$30\)	\(-\$30 + \$100\)
net payoff	\(-\$5\)	\(-\$5\)

Partition ByLength	Partition ByArea
(1) The side length is 1 to 2 cm.	\((1')\) The area is 1 to 2cm².
(2) The side length is 2 to 3 cm.	\((2')\) The area is 2 to 3cm².
(3) The side length is 3 to 4 cm.	\((3')\) The area is 3 to 4cm².
	\(\;\;\vdots\)
	\((15')\) The area is 15 to 16cm²

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