Other Accounts 3.1 Dempster-Shafer Theory

The theory of Dempster-Shafer (DS) belief functions (Dempster 1968, Shafer 1976) rejects the claim that degrees of belief can be measured by the epistemic agent’s betting behavior. A particular version of the theory of DS belief functions is the transferable belief model (Smets & Kennes 1994). It distinguishes between two mental levels: the credal level, where one entertains and quantifies various beliefs, and the pignistic level, where one uses those beliefs for decision making. Its twofold thesis is that (fair) betting ratios should indeed obey the probability calculus, but that degrees of belief, being different from (fair) betting ratios, need not. It suffices that they satisfy the weaker DS principles. The idea is that whenever one is forced to bet on the pignistic level, the degrees of belief from the credal level are used to calculate (fair) betting ratios that satisfy the probability axioms. These in turn are then used to calculate the agent’s expected utility for various acts (Buchak 2014, Joyce 1999, Savage 1972). However, on the credal level degrees of belief need not obey the probability calculus.
Whereas subjective probabilities are additive (axiom 3), DS belief functions Bel: [ltr]A→RA→ℜ[/ltr] are only super-additive, i.e., for all propositions [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]:

[ltr]Bel(A)+Bel(B)≤Bel(A∪B) if A∩B=∅.(6)(6)Bel(A)+Bel(B)≤Bel(A∪B) if A∩B=∅.[/ltr]

In particular, the agent’s degree of belief for [ltr]AA[/ltr] and her degree of belief for [ltr]W∖AW∖A[/ltr] need not sum to 1.
What does it mean to say that Sophia’s degree of belief that tomorrow it will be sunny in Vienna equals .55, if her degrees of belief are represented by a DS belief function [ltr]Bel:A→RBel:A→ℜ[/ltr]? According to one interpretation (Haenni & Lehmann 2003), the number [ltr]Bel(A)Bel(A)[/ltr] represents the strength with which [ltr]AA[/ltr] is supported by the agent’s knowledge or belief base. It may well be that this base neither supports [ltr]AA[/ltr] nor its complement [ltr]W∖AW∖A[/ltr]. Recall the supposition that Sophia has hardly any enological knowledge. Under this assumption her knowledge or belief base will neither support the proposition [ltr]RedRed[/ltr] that a Schilcher is a red wine nor will it support the proposition [ltr]WhiteWhite[/ltr] that a Schilcher is a white wine. However, due to a different aspect of her enological ignorance (namely that she does not know that there are wines, namely rosés, which are neither red nor white), Sophia may well be certain that a Schilcher is either a red wine or a white wine. Hence Sophia’s DS belief function [ltr]BelBel[/ltr] will be such that [ltr]Bel(Red)=Bel(White)=0Bel(Red)=Bel(White)=0[/ltr] while [ltr]Bel(Red∪White)=1Bel(Red∪White)=1[/ltr]. On the other hand, Sophia knows for sure that the coin she is about to toss is fair. Hence her [ltr]BelBel[/ltr] will be such that [ltr]Bel(H)=Bel(T)=.5Bel(H)=Bel(T)=.5[/ltr] as well as [ltr]Bel(H∪T)=1Bel(H∪T)=1[/ltr]. Thus we see that the theory of DS belief functions can distinguish between uncertainty and one form of ignorance. Indeed,

[ltr]I({A[size=13]i})=1−Bel(A1)−…−Bel(An)−…I({Ai})=1−Bel(A1)−…−Bel(An)−…[/ltr][/size]

can be seen as a measure of the agent’s ignorance with respect to the countable partition [ltr]{A[size=13]i}{Ai}[/ltr](the [ltr]AiAi[/ltr] may, for instance, be the values of a random variable such as the price of a bottle of Schilcher in Vienna on Aug 8, 2008).[/size]
Figuratively, a proposition [ltr]AA[/ltr] divides the agent’s knowledge or belief base into three mutually exclusive and jointly exhaustive parts: a part that speaks in favor of [ltr]AA[/ltr], a part that speaks against [ltr]AA[/ltr](i.e., in favor of [ltr]W∖A)W∖A)[/ltr], and a part that neither speaks in favor of nor against [ltr]AA[/ltr]. [ltr]Bel(A)Bel(A)[/ltr]quantifies the part that supports [ltr]A,Bel(W∖A)A,Bel(W∖A)[/ltr] quantifies the part that supports [ltr]W∖AW∖A[/ltr], and I[ltr]({A,W∖A})=1−Bel(A)−Bel(W∖A)({A,W∖A})=1−Bel(A)−Bel(W∖A)[/ltr] quantifies the part that supports neither [ltr]AA[/ltr] nor [ltr]W∖AW∖A[/ltr]. Formally this is spelt out in terms of a (normalized) mass function on [ltr]AA[/ltr], a function m: [ltr]A→RA→ℜ[/ltr] such that for all propositions [ltr]AA[/ltr] in [ltr]AA[/ltr]:

[ltr]m(A)m(∅)∑[size=13]B∈Am(B)≥0,=0 (normalization), and =1.m(A)≥0,m(∅)=0 (normalization), and ∑B∈Am(B)=1.[/ltr][/size]

A (normalized) mass function m: [ltr]A→RA→ℜ[/ltr] induces a DS belief function Bel: [ltr]A→RA→ℜ[/ltr] by defining for each [ltr]AA[/ltr] in [ltr]AA[/ltr],

[ltr]Bel(A)=∑[size=13]B⊆A,B∈Am(B).Bel(A)=∑B⊆A,B∈Am(B).[/ltr][/size]

The relation to subjective probabilities can now be stated as follows. Subjective probabilities require the ideal doxastic agent to divide her knowledge or belief base into two mutually exclusive and jointly exhaustive parts: one that speaks in favor of [ltr]AA[/ltr], and one that speaks against [ltr]AA[/ltr]. That is, the neutral part has to be distributed among the positive and negative parts. Subjective probabilities can thus be seen as DS belief functions without ignorance. (See Pryor (manuscript, Other Internet Resources) for a model of doxastic states that comprises probability theory and Dempster-Shafer theory as special cases.)
A DS belief function [ltr]Bel:A→RBel:A→ℜ[/ltr] induces a Dempster-Shafer plausibility function [ltr]P:A→RP:A→ℜ[/ltr], where for all [ltr]AA[/ltr] in [ltr]AA[/ltr],

[ltr]P(A)=1−Bel(W∖A).P(A)=1−Bel(W∖A).[/ltr]

Degrees of plausibility quantify that part of the agent’s knowledge or belief base which is compatible with [ltr]AA[/ltr], i.e., the part that supports [ltr]AA[/ltr] and the part that supports neither [ltr]AA[/ltr] nor [ltr]W∖AW∖A[/ltr]. In terms of the (normalized) mass function m inducing Bel this means that

[ltr]P(A)=∑[size=13]B∩A≠∅,B∈Am(B).P(A)=∑B∩A≠∅,B∈Am(B).[/ltr][/size]

If, and only if, [ltr]Bel(A)Bel(A)[/ltr] and [ltr]Bel(W∖A)Bel(W∖A)[/ltr] sum to less than 1, [ltr]P(A)P(A)[/ltr] and [ltr]P(W∖A)P(W∖A)[/ltr] sum to more than 1. For an overview see Haenni (2009).
The theory of DS belief functions is more general than the theory of subjective probability in the sense that the latter requires degrees of belief to be additive, while the former merely requires them to be super-additive. In another sense, though, the converse is true. The reason is that DS belief functions can be represented as convex sets of probabilities (Walley 1991). As not every convex set of probabilities can be represented as a DS belief function, sets of probabilities provide the most general framework we have come across so far. An even more general framework is provided by Halpern’s plausibility measures (Halpern 2003). These are functions Pl: [ltr]A→RA→ℜ[/ltr] such that for all [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]:

[ltr]Pl(∅)Pl(W)=0,=1,Pl(∅)=0,Pl(W)=1,[/ltr]

and

[ltr]Pl(A)≤Pl(B) if A⊆B.(7)(7)Pl(A)≤Pl(B) if A⊆B.[/ltr]

In fact, these are only the special cases of real-valued plausibility measures. While it is fairly uncontroversial that an ideal doxastic agent’s degree of belief function should obey Halpern’s plausibility calculus, it is questionable whether his minimal principles are all there is to the rationality of degrees of belief. The resulting epistemology is, in any case, very thin. It should be noted, though, that Halpern does not intend plausibility measures to provide a complete epistemology, but rather a general framework to study more specific accounts

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Other Accounts 3.1 Dempster-Shafer Theory

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Other Accounts 3.1 Dempster-Shafer Theory

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