The theory of subjective probabilities is not an adequate description of people’s doxastic states (Kahneman & Slovic & Tversky 1982). It is a normative theory that tells us how an ideal doxastic agent’s degrees of belief should behave. The thesis that an ideal doxastic agent’s degrees of belief should obey the probability calculus is known as probabilism. So, why should such an agent’s degrees of belief obey the probability calculus?
The Dutch Book Argument provides an answer to this question. (Cox’s theorem, Cox 1946, and the representation theorem of measurement theory, Krantz & Luce & Suppes & Tversky 1971, provide two further answers. For criticism of the latter see Meacham & Weisberg 2011.) On its standard, pragmatic reading, the Dutch Book Argument starts with a link between degrees of belief and betting ratios. The second premise says that it is (pragmatically) defective to accept a series of bets which guarantees a loss. Such a series of bets is called a Dutch Book (hence the name ‘Dutch Book Argument’). The third ingredient is the Dutch Book Theorem. The standard, pragmatic version says that an agent’s betting ratios obey the probability calculus if and only if an agent who has these betting ratios cannot be Dutch Booked (i.e., presented a series of bets each of which is acceptable according to these betting ratios, but their combination guarantees a loss). From this it is inferred that it is (doxastically) defective to have degrees of belief that do not obey the probability calculus. This argument would be valid only if the link between degrees of belief and betting ratios were identity (in which case there would be no difference between pragmatic and doxastic defectiveness) — and we have already seen that it is not.
Hence there is a depragmatized Dutch Book Argument (cf. Armendt 1993, Christensen 1996, Ramsey 1926, Skyrms 1984). From a link between degrees of belief and fair betting ratios and the assumption that it is (doxastically) defective to consider a Dutch Book as fair, it is inferred that it is (doxastically) defective to have degrees of belief that violate the probability calculus. The version of the Dutch Book Theorem that licenses this inference says that an agent’s fair betting ratios obey the probability calculus if and only if the agent never considers a Dutch Book as fair. The depragmatized Dutch Book Argument is a more promising justification for probabilism. See, however, Hájek (2005; 2008).
Joyce (1998) attempts to vindicate probabilism by considering the accuracy of degrees of belief. The basic idea here is that a degree of belief function is (doxastically) defective if there exists an alternative degree of belief function that is more accurate in each possible world. The accuracy of a degree of belief [ltr]b(A)b(A)[/ltr] in a proposition [ltr]AA[/ltr] in a world [ltr]ww[/ltr] is identified with the distance between [ltr]b(A)b(A)[/ltr] and the truth value of [ltr]AA[/ltr] in [ltr]ww[/ltr], where 1 represents truth and 0 represents falsity. For instance, a degree of belief up to 1 in a true proposition is more accurate, the higher it is — and perfectly accurate if it equals 1. The overall accuracy of a degree of belief function [ltr]bb[/ltr] in a world [ltr]ww[/ltr]is then determined by the accuracy of the individual degrees of belief [ltr]b(A)b(A)[/ltr]. Joyce is able to prove that, given some conditions on how to measure distance or inaccuracy, a degree of belief function obeys the probability calculus if and only if there exists no alternative degree of belief function that is more accurate in each possible world (the only-if-part is not explicitly mentioned in Joyce 1998, but needed for the argument to work and present in Joyce 2009). Therefore, degrees of belief should obey the probability calculus.
The objection, known as Bronfman’s objection, that has attracted most attention starts by noting that Joyce’s conditions on measures of inaccuracy do not determine a single measure, but rather a whole set of such measures. This would strengthen rather than weaken Joyce’s argument, were it not for the fact that these measures differ in their recommendations as to which alternative degree of belief function a non-probabilistic degree of belief function should be replaced by. All of Joyce’s measures of inaccuracy agree that an agent whose degree of belief function violates the probability axioms should adopt a probabilistic degree of belief function which is more accurate in each possible world. However, these measures may differ in their recommendation as to which particular probability measure the agent should adopt. In fact, for each possible world, following the recommendation of one measure will leave the agent off less accurate according to some other measure. Why, then, should the ideal doxastic agent move from her non-probabilistic degree of belief function to a probability measure in the first place? Other objections are articulated in Maher (2002) and, more recently, in Easwaran & Fitelson (2012) (see, however, the replies by Joyce ms (Other Internet Resources) and Pettigrew 2013). Joyce (2009) responds to some of these objections. Leitgeb & Pettigrew (2010a; 2010b) present conditions that narrow down the set of measures of inaccuracy to the so-called quadratic scoring rules of the form [ltr]λ(b(A)−w(A))[size=13]2λ(b(A)−w(A))2[/ltr], where [ltr]w(A)w(A)[/ltr] is 1 if [ltr]AA[/ltr] is true in [ltr]ww[/ltr], and 0 otherwise. This enables them to escape Bronfman’s objection.[/size]

2.4 Update Rules

We have discussed how to measure and interpret subjective probabilities, and why degrees of belief should be subjective probabilities. It is of particular epistemological interest how to update subjective probabilities when new information is received. Whereas axioms 1–5 of the probability calculus are synchronic conditions on an agent’s degree of belief function, update rules are diachronic conditions that tell us how to revise our subjective probabilities when we receive new information of a certain format. If the new information comes in form of a certainty, probabilism is extended by
Strict Conditionalization 
If evidence comes only in the form of certainties (that is, propositions of which you become certain), if Pr: [ltr]A→RA→ℜ[/ltr] is your subjective probability at time [ltr]tt[/ltr], and if between [ltr]tt[/ltr] and [ltr]t[size=13]′t′[/ltr] you become certain of [ltr]A∈AA∈A[/ltr] and no logically stronger proposition in the sense that your new subjective probability for [ltr]AA[/ltr], but for no logically stronger proposition, is 1 (and your subjective probabilities are not directly affected in any other way such as forgetting etc.), then your subjective probability at time [ltr]t′t′[/ltr] should be [ltr]Pr(⋅∣A)Pr(⋅∣A)[/ltr].[/size]
Strict conditionalization thus says that the agent’s new subjective probability for a proposition [ltr]BB[/ltr]after becoming certain of [ltr]AA[/ltr] should equal her old subjective probability for [ltr]BB[/ltr] conditional on [ltr]AA[/ltr].
Two questions arise. First, why should we update our subjective probabilities according to strict conditionalization? Second, how should we update our subjective probabilities when the new information is of a different format and we do not become certain of a proposition, but merely change our subjective probabilities for various propositions? Jeffrey (1983a) answers the second question by what is now known as Jeffrey conditionalization. The propositions whose (non-conditional) probabilities change as a result of the evidential experience are called evidential propositions. Roughly, Jeffrey conditionalization says that the ideal doxastic agent should keep fixed her “inferential beliefs,” that is, the probabilities of all hypotheses conditional on any evidential proposition.
Jeffrey Conditionalization 
If evidence comes only in form of new degrees of belief for the elements of a partition, if Pr: [ltr]A→RA→ℜ[/ltr] is your subjective probability at time [ltr]tt[/ltr], and if between [ltr]tt[/ltr] and [ltr]t[size=13]′t′[/ltr] your subjective probabilities in the mutually exclusive and jointly exhaustive propositions [ltr]Ai∈AAi∈A[/ltr] are directly affected and change to [ltr]pi∈[0,1]pi∈[0,1][/ltr] with [ltr]∑ipi=1∑ipi=1[/ltr], and the positive part of your subjective probability is not directly affected on any superset of the partition [ltr]{Ai}{Ai}[/ltr] (and your subjective probabilities are not directly affected in any other way such as forgetting etc.), then your subjective probability at time [ltr]t′t′[/ltr] should be [ltr]Pr′(⋅)=∑iPr(⋅∣Ai)piPr′(⋅)=∑iPr(⋅∣Ai)pi[/ltr].[/size]
Jeffrey conditionalization thus says that the agent’s new subjective probability for [ltr]BB[/ltr], after her subjective probabilities for the elements [ltr]A[size=13]iAi[/ltr] of a partition have changed to [ltr]pipi[/ltr], should equal the weighted sum of her old subjective probabilities for [ltr]BB[/ltr] conditional on the [ltr]AiAi[/ltr], where the weights are the new subjective probabilities [ltr]pipi[/ltr] for the elements of the partition.[/size]
One answer to the first question is the Lewis-Teller Dutch Book Argument (Lewis 1999, Teller 1973). Its extension to Jeffrey conditionalization is presented in Armendt (1980) and discussed in Skyrms (1987). For more on diachronic coherence see Skyrms (2006). Leitgeb & Pettigrew (2010b) present a gradational accuracy argument for strict conditionalization (see also Greaves & Wallace 2006) as well as an argument for an alternative to Jeffrey conditionalization (for an overview see the excellent entry on epistemic utility arguments for probabilism). Other philosophers have provided arguments against strict (and Jeffrey) conditionalization: van Fraassen (1989) holds that rationality does not require the adoption of a particular update rule (but see Hájek 1998 and Kvanvig 1994). Arntzenius (2003) uses, among others, the “shifting” nature of self-locating beliefs to argue against strict conditionalization as well as against van Fraassen’sreflection principle (van Fraassen 1995; for an illuminating discussion of the reflection principle and Dutch Book arguments see Briggs 2009a). The second feature used by Arntzenius (2003), called “spreading”, is not special to self-locating beliefs. Weisberg (2009) argues that Jeffrey conditionalization cannot handle a phenomenon he terms perceptual undermining.