aa r X i v : . [ m a t h . S T ] J u l On Accuracy and Coherence withInfinite Opinion Sets
Mikayla Kelley
Stanford University ([email protected])
Preprint of July 2020 (under review)
Abstract
There is a well-known equivalence between avoiding accuracy dominance and having prob-abilistically coherent credences (see, e.g., de Finetti 1974, Joyce 2009, Predd et al. 2009,Schervish et al. 2009, Pettigrew 2016). However, this equivalence has been established onlywhen the set of propositions on which credence functions are defined is finite. In this paper, weestablish connections between accuracy dominance and coherence when credence functions aredefined on an infinite set of propositions. In particular, we establish the necessary results toextend the classic accuracy argument for probabilism originally due to Joyce (1998) to certainclasses of infinite sets of propositions including countably infinite partitions.
A central norm in the epistemology of partial belief is probabilism: a person’s degrees of belief—or credences —should satisfy the laws of probability. There is a long tradition in the spirit of Savage(1971) and de Finetti (1974) of appealing to the epistemic virtue of accuracy to justify probabilism(also see Rosenkrantz 1981). One particular form of argument is the accuracy dominance argumentfor probabilism introduced by Joyce (1998). Let a set F of propositions be an opinion set and afunction c : F → [0 , a credence function on F . Let a credence function be coherent if it satisfiesthe axioms of probability. A credence function c ′ on F accuracy dominates a credence function c on F if c is more inaccurate than c ′ no matter how the world turns out to be (where inaccuracy isprecisified as in Section 2). Then the existing accuracy dominance arguments purport to vindicateprobabilism by showing that a credence function is not accuracy dominated if and only if it iscoherent.However, there is a limitation to almost all of the literature on accuracy arguments for proba-bilism: the opinion set is assumed to be finite. Indeed, de Finetti (1974), Lindley (1987), Joyce This paper is based on work done in Kelley 2019. In an unpublished manuscript, Walsh (2020) gives an accuracy dominance argument in the countably infinitecontext, to which we return in Section 3. In a related but distinct area, Huttegger (2013) and Easwaran (2013)extend to the infinite setting part of the literature on using minimization of expected inaccuracy to vindicateepistemic principles. See, e.g., Greaves and Wallace 2005. Schervish et al. (2014) prove that in certain countablyinfinite cases, coherence is sufficient to avoid strong dominance . Schervish et al. (2009) and Steeger (2019) explorea different way to weaken the assumption that the opinion set is finite. We return to their work in Section 4.
We first set up the framework that will be used throughout the paper. Fix a set W (not necessarilyfinite) which represents the set of possible worlds and, for now, a finite set F ⊆ P ( W ) of propositions that represents an opinion set —the set of propositions that an agent has beliefs about. Definition 2.1. An algebra over W is a subset F ∗ ⊆ P ( W ) such that:1. W ∈ F ∗ ;2. if p, p ′ ∈ F ∗ , then p ∪ p ′ ∈ F ∗ ;3. if p ∈ F ∗ , then W \ p ∈ F ∗ . Definition 2.2. i. A credence function on an opinion set F is a function from F to [0 , .ii. A credence function c is coherent if it can be extended to a finitely additive probabilityfunction on an algebra F ∗ over W containing F . That is, there is an algebra F ∗ ⊇ F over W and a function c ∗ : F ∗ → [0 , such that:(a) c ∗ ( p ) = c ( p ) for all p ∈ F ;(b) c ∗ ( p ∪ p ′ ) = c ∗ ( p ) + c ∗ ( p ′ ) for p, p ′ ∈ F ∗ with p ∩ p ′ = ∅ ;(c) c ∗ ( W ) = 1 .iii. A credence function that is not coherent is incoherent . Remark 2.3. If F = { p , . . . , p n } , we identify a credence function c over F with the vector ( c ( p ) , . . . , c ( p n )) ∈ [0 , n . Thus the space of all credence functions over F can be identified with [0 , n ⊆ R n . We often simplify notation by setting c i := c ( p i ) .We now introduce an important subclass of the class of all credence functions, namely the(coherent) credence functions that match the truth values of F at a world w exactly. Definition 2.4.
Fix an opinion set F . For each w ∈ W , let v w : F → { , } be defined by v w ( p ) = 1 if and only if w ∈ p . We call v w the omniscient credence function at world w . We let V F denote the set of all omniscient credence functions on F . Note that |V F | ≤ |F| .2ext, we specify the inaccuracy measures we will be concerned with in this section. Fix afinite opinion set F , and let C denote the set of credence functions on F . We define an inaccuracymeasure to be a function of the form I : C × W → [0 , ∞ ] . The class of inaccuracy measures we consider is a generalization of the class normatively defendedby Pettigrew (2016): the inaccuracy measures defined in terms of what we call a quasi-additiveBregman divergence . It is a subclass of the inaccuracy measures assumed in Predd et al. 2009. Definition 2.5.
Suppose D : [0 , n × [0 , n → [0 , ∞ ] .1. D is a divergence if D ( x , y ) ≥ for all x , y ∈ [0 , n with equality if and only if x = y .2. D is quasi-additive if there exists a function d : [0 , → [0 , ∞ ] and a sequence of elements { a i } ni =1 from (0 , ∞ ) such that D ( x , y ) = n X i =1 a i d ( x i , y i ) , in which case we say D is generated by d and { a i } ni =1 .3. D is a quasi-additive Bregman divergence if D is a quasi-additive divergence generated by d and { a i } ni =1 , and in addition there is a function ϕ : [0 , → R such that:(a) ϕ is continuous, bounded, and strictly convex on [0 , ;(b) ϕ is continuously differentiable on (0 , with the formal definition ϕ ′ ( i ) := lim x → i ϕ ′ ( x ) for i ∈ { , } ; (c) for all x, y ∈ [0 , , we have d ( x, y ) = ϕ ( x ) − ϕ ( y ) − ϕ ′ ( y )( x − y ) . We call such a d a one-dimensional Bregman divergence .We take the inaccuracy of a credence function c at a world w to be the distance between c andthe omniscient credence function v w , where distance is measured with a quasi-additive Bregmandivergence. Definition 2.6.
Let a legitimate inaccuracy measure be an inaccuracy measure given by I ( c, w ) = D ( v w , c ) , where D is a quasi-additive Bregman divergence. Using terminology from Definition 2.5, Predd et al. consider a more general class in allowing different one-dimensional Bregman divergences for different propositions. We do not require ϕ ′ ( i ) < ∞ for i ∈ { , } .
3y allowing different weights depending on the proposition, we can accommodate the intuitionthat some propositions are more important to know than others (see, e.g., Levinstein 2019 forfurther discussion of the varying epistemic importance of propositions). Even if one thinks thatinaccuracy measures should be additive, as Pettigrew (2016) does, relaxing this restriction makesour results more widely relevant. A popular example of an additive legitimate inaccuracy measureis the Brier score (see Section 12, “Homage to the Brier Score,” of Joyce 2009): I ( c, w ) = n X i =1 ( v w ( p i ) − c ( p i )) . Remark 2.7.
The class of additive Bregman divergences is the class of additive and continuous strictly proper scoring rules . See Pettigrew 2016, p. 66. Also see, e.g., Banerjee et al. 2005 andGneiting and Raftery 2007 for more details on Bregman divergences as well as their connection tostrictly proper scoring rules.We now recall the dominance result connecting coherence to accuracy dominance when theopinion set is finite. It was first proved for the Brier score by de Finetti (1974, pp. 87-90) andextended to any legitimate inaccuracy measure by Predd et al. (2009). See Schervish et al. 2009for further generalizations of the finite result.
Definition 2.8.
For each pair of credence functions c, c ∗ over F :1. c ∗ weakly dominates c relative to an inaccuracy measure I if I ( c, w ) ≥ I ( c ∗ , w ) for all w ∈ W and I ( c, w ) > I ( c ∗ , w ) for some w ∈ W ;2. c ∗ strongly dominates c relative to I if I ( c, w ) > I ( c ∗ , w ) for all w ∈ W . Theorem 2.9 (de Finetti 1974, Predd et al. 2009) . Let F be a finite opinion set, I a legitimateinaccuracy measure, and c a credence function on F . Then the following are equivalent:1. c is not strongly dominated;2. c is not weakly dominated;3. c is coherent.Further, if c is incoherent, then c is strongly dominated by a coherent credence function.On the basis of Theorem 2.9, authors in the accuracy literature conclude that an incoherentcredence function is objectionable because there is an undominated coherent credence functionthat does strictly better in terms of accuracy, no matter how the world turns out to be, whereascoherent credence functions are not accuracy dominated in this way. Since it is the basis of theaccuracy argument for probabilism in the finite case, Theorem 2.9 is the result we would like toextend to infinite opinion sets. We now make progress toward this goal when F is countablyinfinite. 4 The Countable Case: Coherence is Necessary
We begin with a discussion of how to measure inaccuracy in the countably infinite setting. Fix acountably infinite opinion set F over a set W of worlds (of arbitrary cardinality). Let C be the setof credence functions over F , which can be identified with [0 , ∞ (see Remark 2.3). An inaccuracymeasure remains a map from C × W to [0 , ∞ ] .The class of inaccuracy measures that we use are defined in terms of generalizations of quasi-additive Bregman divergences. Definition 3.1.
Suppose D : [0 , ∞ × [0 , ∞ → [0 , ∞ ] . Then we call D a generalized quasi-additiveBregman divergence if D ( x , y ) = ∞ X i =1 a i d ( x i , y i ) , where d is a bounded one-dimensional Bregman divergence as in Definition 2.5.3 and { a i } ∞ i =1 asequence of elements from (0 , ∞ ) with sup i a i < ∞ . Remark 3.2.
Note that d —defined in terms of ϕ —being bounded is equivalent to ϕ ′ beingbounded on [0 , . Further, we may assume that ϕ (0) = ϕ ′ (0) = 0 since d ϕ = d ¯ ϕ if ϕ and ¯ ϕ differ by a linear function. In the appendix, we show that generalized quasi-additive Bregman divergences are examples ofwhat Csiszár (1995) calls
Bregman distances , which are generalizations of quasi-additive Bregmandivergences defined on spaces of non-negative functions.Suggestively, we make the following definition.
Definition 3.3.
Given an enumeration of F , let a generalized legitimate inaccuracy measure bean inaccuracy measure I : C × W → [0 , ∞ ] given by I ( c, w ) = D ( v w , c ) (1)for D a generalized quasi-additive Bregman divergence.Notice that the Brier score extends to a generalized legitimate inaccuracy measure, namely thesquared ℓ ( F ) norm I ( c, w ) = || v w − c || ℓ ( F ) = ∞ X i =1 ( v w ( p i ) − c ( p i )) . (2)We call (2) the generalized Brier score .The name “generalized legitimate inaccuracy measure” is motivated by the observation thata generalized legitimate inaccuracy measure naturally restricted to the finite opinion sets is a le-gitimate inaccuracy measure. This is because 1) for both the generalized and finite legitimate Recall that sup i a i = a ∈ R ∪ { + ∞ , −∞} such that a i ≤ a for all i ∈ N and for any b < a , there is some a i suchthat b < a i ≤ a . Proof: Let ¯ ϕ ( x ) = ϕ ( x ) + ax + b . Then d ¯ ϕ ( x, y ) = ϕ ( x ) + ax + b − ϕ ( y ) − ay − b − ( ϕ ′ ( y ) + a )( x − y ) = ϕ ( x ) + ax + b − ϕ ( y ) − ay − b − ϕ ′ ( y )( x − y ) − ax + ay = ϕ ( x ) − ϕ ( y ) − ϕ ′ ( y )( x − y ) = d ϕ ( x, y ) . Further, if ϕ satisfiesthe conditions in Definition 2.5.3, then ¯ ϕ does as well. Thus we may assume that any one-dimensional Bregmandivergence is defined by a function ϕ such that ϕ (0) = ϕ ′ (0) = 0 . The choice of enumeration does not matter since the terms in the infinite sum defining inaccuracy are non-negative. Thus convergence is absolute and independent of order.
We now state one of our main results: coherence is necessary to avoid accuracy dominance in thecountably infinite case.
Theorem 3.4.
Let F be a countably infinite opinion set, I a generalized legitimate inaccuracymeasure, and c an incoherent credence function. Then:1. c is weakly dominated relative to I by a coherent credence function; and2. if I ( c, w ) < ∞ for each w ∈ W , then c is strongly dominated relative to I by a coherentcredence function. Proof.
See the Appendix.
Remark 3.5.
By analyzing the proof of Theorem 3.4, one can see that the most general way tostate the theorem is: assume c is incoherent; if I ( c, w ) < ∞ for some w , then there is a coherentcredence function d such that I ( d, w ) < I ( c, w ) for all w such that I ( c, w ) < ∞ ; if I ( c, w ) = ∞ for all w ∈ W , then any omniscient credence function weakly dominates c . Remark 3.6.
The following is easy to prove from the results of Schervish et al. (2009): anyincoherent credence function c over a countably infinite opinion set is weakly dominated but notnecessarily by a coherent credence function; and if I ( c, w ) < ∞ for each w ∈ W , then c isstrongly dominated but not necessarily by a coherent credence function. Thus the value in theproof strategy to come is that the dominating credence function is proven to be coherent, whichis analogous to the finite case.
We note that one direction of Walsh’s (2020) accuracy dominance result follows immediatelyfrom Theorem 3.4. We first recall his result. Proof sketch: If c is incoherent, then there is some finite F ⊆ F on which c is incoherent. Restrict c to c on F . Then by Theorem 2.9, there is some d that strongly dominates c . Extend d to a credence function d on F bycopying c off of F . Then so long as c has finite inaccuracy at some world, d will weakly dominate c . Thanks to Teddy Seidenfeld for suggesting this connection to the finite case. Further, it is often argued that not all dominated credence functions are irrational—only those that are domi-nated by a credence function which is itself not dominated (see discussion of the Undominated Dominance principlein Pettigrew 2016, p. 22). For the opinion sets and inaccuracy measures dicussed in Section 4, the undominatedcredence functions will be precisely the coherent credence functions, and so the added strength of Theorem 3.4 isnormatively important, as well. heorem 3.7 (Walsh 2020) . Let F be a countably infinite opinion set. Let I ( c, w ) = ∞ X i =1 − i ( v w ( p i ) − c ( p i )) . (3)Then:1. if c is incoherent, then c is strongly dominated relative to I by a coherent credence function;2. if c is coherent, then c is not weakly dominated relative to I by any credence function d = c .Part 1 of this result follows from Theorem 3.4 by defining I in terms of the generalized quasi-additive Bregman divergence generated by { − i } ∞ i =1 and d ( x, y ) = x − y − xy ( x − y ) = ϕ ( x ) − ϕ ( y ) − ϕ ′ ( y )( x − y ) , where ϕ ( x ) = x . Note that I ( c, w ) < ∞ for all c ∈ C and w ∈ W as P i − i < ∞ . Unlike coherent credence functions on finite opinion sets, coherent credence functions on countablyinfinite opinion sets can be strongly dominated.
Example 4.1.
Let F = {{ n ≥ N : n ∈ N } : N ∈ N } be an opinion set over N (including zero).Let c ( { n ≥ N } ) = 1 √ N + 1 . Then c is coherent—in fact, countably coherent (see Definition 4.6)—but I ( c, w ) = ∞ for all w ∈ W when I is the generalized Brier score. So any omniscient credence function stronglydominates c .In fact, the classic example of a merely finitely additive probability function—the 0-1 functiondefined on the finite-cofinite algebra over N taking value 0 on finite sets—restricts to a coherentdominated credence function. Example 4.2.
Let F = {{ n ≤ N : n ∈ N } : N ∈ N } be an opinion set over N (including zero).Let c ( { n ≤ N } ) = 0 . Then c is coherent—as well as finitely supported and not countably coherent—but I ( c, w ) = ∞ for all w ∈ W when I is the generalized Brier score. So any omniscient credence function stronglydominates c .The goal of this section is to characterize the opinion sets and inaccuracy measures for whichsome variant of Theorem 2.9 holds. We extend Theorem 2.9 by proving dominance results for countably coherent credence functions and using an opinion set compactification construction totransfer these results to merely coherent credence functions. At points, our results will only applyto the generalized Brier score. We conjecture that any such result extends to any generalizedlegitimate inaccuracy measure. In any case, this is a well motivated restriction since the Brier7core has been defended by many—including Horwich (1982), Maher (2002), Joyce (2009), andLeitgeb and Pettigrew (2010a)—as being a particularly appropriate way to measure inaccuracy. Throughout the rest of this section we assume the opinion set F is countably infinite. We begin by introducing the notion of a countably coherent credence function and establishing acharacterization theorem regarding countable coherence on countably discriminating opinion setswhich extends a result of de Finetti (1974).
Definition 4.3.
For
F ⊆ P ( W ) , we define an equivalence relation ∼ on W such that w ∼ w ′ ifand only if { p ∈ F : w ∈ p } = { p ∈ F : w ′ ∈ p } . We call the set of equivalence classes of W the quotient of W relative to F . If the quotient of W relative to F is countable, then we call F countably discriminating .Clearly, any countable opinion set over a countable set of worlds is countably discriminating.The following characterization of the coherent credence functions on finite opinion sets is dueto de Finetti (1974). Recall V F is the set of omniscient credence functions on F , which is finitewhen F is finite. Theorem 4.4 (de Finetti 1974) . c is a coherent credence function on a finite opinion set F if andonly if there are λ w ∈ [0 , with P v w ∈V F λ w = 1 such that c ( p ) = X v w ∈V F λ w v w ( p ) for all p ∈ F .Theorem 4.4 is integral to Predd et al.’s proof that coherence is sufficient to avoid dominancein Theorem 2.9. We now show de Finetti’s characterization of the coherent credence functions onfinite opinion sets extends to countably coherent credence functions on countably discriminatingopinion sets. Definition 4.5. A σ -algebra over W is a subset F ∗ ⊆ P ( W ) such that:1. W ∈ F ∗ ;2. if { p i } ∞ i =1 ⊆ F ∗ , then S ∞ i =1 p i ∈ F ∗ ;3. if p ∈ F ∗ , then W \ p ∈ F ∗ . Definition 4.6.
Let a credence function c be countably coherent if c extends to a countablyadditive probability function on a σ -algebra F ∗ containing F . That is, there is a c ∗ : F ∗ → [0 , such that:1. c ∗ ( p ) = c ( p ) for all p ∈ F ;2. c ∗ ( S ∞ i =1 p i ) = P ∞ i =1 c ∗ ( p i ) for { p i } ∞ i =1 ⊆ F ∗ with p i ∩ p j = ∅ for i = j ; Note that if c is countably coherent on F , then c extends to a countably additive probability function on σ ( F ) ,the σ -algebra generated by F . c ∗ ( W ) = 1 .Otherwise, a credence function is countably incoherent . Proposition 4.7.
Let F be a countably discriminating opinion set. Then a credence function c is countably coherent if and only if there are λ w ∈ [0 , with P v w ∈V F λ w = 1 such that c ( p ) = X v w ∈V F λ w v w ( p ) for all p ∈ F . Proof.
See the Appendix.
In this section, we introduce the compactification construction of what we call an opinion space .The construction will be relevant to transferring dominance results for countably coherent credencefunctions to merely coherent credence functions.
Definition 4.8. An opinion space is a pair ( W, F ) , where W is a nonempty set and F ⊆ P ( W ) .From here on out we will speak in terms of opinion spaces as opposed to opinion sets in order tokeep track of the underlying set of worlds.Borkar et al. (2003) proved that the opinion spaces which satisfy a certain compactness propertyare precisely those where the set of coherent credence functions and the set of countably coherentcredence functions coincide. Definition 4.9.
Let ( W, F ) be an opinion space. Let f ( n ) ∈ { , } and set p f ( n ) n = p n if f ( n ) = 0 and p f ( n ) n = p cn if f ( n ) = 1 . Then ( W, F ) is compact if for any choice of { p n } ∞ n =1 ⊆ F and f : N → { , } , if T Nn =1 p f ( n ) n is nonempty for every N , then T ∞ n =1 p f ( n ) n is nonempty.As an example, note that the opinion spaces from Examples 4.1 and 4.2 are not compact. In-deed, for the first example T ∞ n =1 p n = ∅ and yet every finite subset of F has nonempty intersection;for the second example, T ∞ n =1 p cn = ∅ while T Nn =1 p cn = ∅ for every N . Remark 4.10.
Assume F is closed under finite intersections. Let A ( F ) denote the algebragenerated by F , and let T ( F ) denote the topology generated by F . By the Alexander subbasetheorem (see, e.g., Kelley 1975, p. 139), ( W, F ) is compact if and only if T ( A ( F )) is compact. Theorem 4.11 (Borkar et al. 2003) . The following are equivalent:1. ( W, F ) is compact;2. for every credence function c on ( W, F ) , c is coherent if and only if c is countably coherent.We now show how to associate a compact space to any space and, in light of Theorem 4.11, acountably coherent credence function to any coherent credence function. Let ( W, F ) be an opinionspace. Let S denote the set of sequences of the form { p f ( n ) n } (as in Definition 4.9) such that Proof: The set of elements in F and their complements form a subbase for T ( A ( F )) . Nn =1 p f ( n ) n = ∅ for every N but T ∞ n =1 p f ( n ) n = ∅ . Define W ∗ = W ∪ { x s : s ∈ S } . Define F ∗ ⊆ P ( W ∗ ) as follows: for each p ∈ F , let S p denote the set of sequences s of the form { p f ( n ) n } (as in Definition 4.9) such that s ∈ S , p n = p for some n , and f ( n ) = 0 . Then define p ∗ = p ∪ { x s : s ∈ S p } . Finally, let F ∗ = { p ∗ : p ∈ F} . We call ( W ∗ , F ∗ ) the compactification of ( W, F ) . We alwaysdenote the compactification of ( W, F ) by ( W ∗ , F ∗ ) . Further, we let Ψ denote the natural bijectionfrom F to F ∗ given by Ψ( p ) = p ∗ .We first note that ( W ∗ , F ∗ ) is in fact compact. Lemma 4.12.
For any opinion space ( W, F ) , ( W ∗ , F ∗ ) is compact. Proof.
See the Appendix.Next we note that we can naturally identify a coherent credence function on ( W, F ) with a count-ably coherent credence function on ( W ∗ , F ∗ ) . Lemma 4.13.
Let ( W, F ) be an opinion space and c a coherent credence function on ( W, F ) . Let ( W ∗ , F ∗ ) be the compactification of ( W, F ) and define c ∗ (Ψ( p )) := c ( p ) for each p ∈ F . Then c ∗ is a countably coherent credence function on ( W ∗ , F ∗ ) and I ( c, w ) = I ( c ∗ , w ) for w ∈ W . Proof.
See the Appendix.For a coherent credence function c defined on an opinion space ( W, F ) , we let c ∗ denote thecountably coherent credence function on ( W ∗ , F ∗ ) given as in Lemma 4.13. Example 4.14.
As an example, let us compute the compactification of the opinion space fromExample 4.2 and show how to identify a coherent credence function on the space with a countablycoherent credence function on its compactification. We note that only for f ( n ) = 1 for all n ∈ N is T Nn =1 p f ( n ) n nonempty for every N while T ∞ n =1 p f ( n ) n = ∅ . Indeed, assume f ( m ) = 0 for some m . If f ( i ) = 1 for some i ≥ m + 1 , then since p ci ∩ p m = ∅ , we have T in =1 p f ( n ) n = ∅ . So f ( i ) = 0 for all i ≥ m + 1 . But then since T mn =1 p f ( n ) n = ∅ and T mn =1 p f ( n ) n ⊆ p i for all i ≥ m + 1 , it also follows that T ∞ n =1 p f ( n ) n = ∅ , which contradicts our assumption. So S is a single point x , W ∗ = W ∪ { x } , and F ∗ = {{ n ≤ N } : N ∈ N } . F ∗ is identical to F , except that there is a point in the complement ofevery proposition in F ∗ . For a coherent credence function c on ( W, F ) , c ∗ on ( W ∗ , F ∗ ) is identicalto c and is a countably coherent credence function on the compact opinion space ( W ∗ , F ∗ ) . Forexample, for credence function c in Example 4.2, c ∗ extends to the countably additive omniscientcredence function v x on the σ -algebra generated by F ∗ .Using Theorem 4.11, Lemma 4.12 and Lemma 4.13, the proof strategy for extending Theorem2.9 is more precisely as follows. First, we establish dominance results for countably coherent cre-dence functions. Second, we associate each coherent credence function on ( W, F ) with a countablycoherent credence function on ( W ∗ , F ∗ ) as in Lemma 4.13. Lastly, we use the dominance resultsfor countably coherent credence functions to establish dominance results for coherent credencefunctions in certain cases where there is “accuracy dominance stability” in compactifying.10 .3 W-Stable Opinion Spaces In this section, we establish the equivalence between coherence and avoiding weak dominancefor certain opinion spaces (Theorem 4.19), as well as additional results extending Theorem 2.9(Corollary 4.22 and Theorem 4.23). We first note that under certain circumstances countablycoherent credence functions are not weakly dominated (Proposition 4.16 and Proposition 4.17);then we use the compactification construction from the previous section and a property of anopinion space—
W-stability (Definition 4.18)—to establish that for certain opinion spaces, merecoherence is also sufficient to avoid weak dominance.We first prove that if a countably coherent credence function c has finite expected inaccuracy,then c is not weakly dominated. Definition 4.15.
For c a countably coherent credence function and I a generalized legitimateinaccuracy measure, we say that c has finite expected inaccuracy relative to I if c has a countablyadditive extension ¯ c defined on the opinion space ( W, σ ( F )) such that E ¯ c I ( c, · ) < ∞ . For c acoherent but not countably coherent credence function and I a generalized legitimate inaccu-racy measure, we say that c has finite expected inaccuracy relative to I if c ∗ has finite expectedinaccuracy relative to I . Note that it follows by Definition 4.15 that any coherent credence function c has finite expectedinaccuracy if and only if c ∗ has finite expected inaccuracy. Proposition 4.16.
Let ( W, F ) be an opinion space with F countably infinite and I a generalizedlegitimate inaccuracy measure. If c is a countably coherent credence function with finite expectedinaccuracy, then c is not weakly dominated. Proof.
See the Appendix.Here is another dominance result for countably coherent credence functions where we assume F is point-finite ( |{ p ∈ F : w ∈ p }| < ∞ for all w ∈ W ) but weaken the assumption that c hasfinite expected inaccuracy considerably, namely to somewhere finitely inaccurate (there is a w ∈ W such that I ( c, w ) < ∞ ). We also restrict to the generalized Brier score B . Proposition 4.17.
Let ( W, F ) be a point-finite opinion space with F countably infinite and I a generalized legitimate inaccuracy measure. If a credence function c is countably coherent andsomewhere finitely inaccurate relative to B , then c is not weakly dominated relative to B . Proof.
See the Appendix.We now introduce the notion of
W-stability which will allow us to use Propositions 4.16 and4.17 to prove extensions of Theorem 2.9.
Definition 4.18.
Let ( W, F ) be W-stable relative to I if for any coherent credence function c on ( W, F ) , if c is weakly dominated relative to I , then c ∗ on ( W ∗ , F ∗ ) is weakly dominated relativeto I . Consider the measure space ( W, σ ( F ) , µ ) . Note d ( d i , v w ( p i )) = 1 p i ( w ) d (1 , d i ) + (1 − p i ( w )) d (0 , d i ) so that eachterm in I ( d, · ) is measurable for any credence function d , and so the infinite sum is measurable as the finite sumand limit of measurable functions are measurable. Thus we can take the expectation of I ( d, · ) with respect to µ for any credence function d . Theorem 4.19.
Let I be a generalized legitimate inaccuracy measure and ( W, F ) a W-stableopinion space relative to I where all coherent credence functions have finite expected inaccuracyrelative to I . Then the following are equivalent:1. c is coherent;2. c is not weakly dominated. Proof.
We prove that if c is coherent, then c is not weakly dominated. Let ( W ∗ , F ∗ ) be thecompactification of ( W, F ) . If c is coherent on ( W, F ) , then c ∗ is countably coherent by Lemma 4.13.Further c ∗ has finite expected inaccuracy by definition and the assumption that c has finite expectedinaccuracy. So by Proposition 4.16, c ∗ is not weakly dominated. But since ( W, F ) is W-stable thisimplies that c is not weakly dominated. The other direction follows from Theorem 3.4. Remark 4.20.
It is trivial to see that W-stability is necessary for the equivalence of coherenceand not being weakly dominated. It is open how far finite expected inaccuracy can be weakened.
Remark 4.21. If I is defined with summable weights, that is, { a i } ∞ i =1 such that P ∞ i =1 a i < ∞ ,then there is a C < ∞ such that I ( c, w ) < C for all credence functions c and w ∈ W . So, inparticular, all coherent credence functions have finite expected inaccuracy relative to I .If we add in an additional finiteness assumption, then we get the full equivalence of Theorem 2.9. Corollary 4.22.
In Theorem 4.19, if in addition all coherent credence functions c have I ( c, w ) < ∞ for all w ∈ W , then the following are equivalent:1. c is coherent;2. c is not weakly dominated;3. c is not strongly dominated.We combine W-stability and Proposition 4.17 to get another set of sufficient conditions on ( W, F ) for Theorem 2.9 to go through for the generalized Brier score. Theorem 4.23.
Let ( W, F ) be a W-stable opinion space with ( W ∗ , F ∗ ) point-finite such that allcoherent credence functions are somewhere finitely inaccurate relative to B . Then the followingare equivalent:1. c is coherent;2. c is not weakly dominated relative to B ;3. c is not strongly dominated relative to B . Proof. If c is coherent, then c ∗ is countably coherent on a point-finite opinion set. Further, c ∗ issomewhere finitely inaccurate relative to B , as c is somewhere finitely inaccurate by assumption.Thus by Proposition 4.17, c ∗ is not weakly dominated relative to B . By W-stability, c is notweakly dominated relative to B . Clearly if c is not weakly dominated then c is not strongly12ominated. Finally, we show that if c is incoherent then c is strongly dominated. First, if c is notsomewhere finitely inaccurate, then any omniscient credence function strongly dominates c since I ( v w , w ′ ) < ∞ for every w, w ′ ∈ W by point-finiteness. If c is somewhere finitely inaccurate then I ( c, w ) < ∞ for all w ∈ W by point-finiteness. Thus Theorem 3.4 establishes that c is stronglydominated relative to B . Remark 4.24.
We can drop the assumption that all coherent credence functions are somewherefinitely inaccurate in Theorem 4.23 if we strengthen W-stable to compact so that ( W, F ) =( W ∗ , F ∗ ) . Indeed, compactness alongside point-finiteness implies coherent credence functions on ( W, F ) = ( W ∗ , F ∗ ) are somewhere finitely inaccurate: if there were a coherent (and thus countablycoherent) credence function infinitely inaccurate at all worlds, then it would be strongly dominatedby an omniscient credence function, contradicting Proposition 4.27 below. As an application of Theorem 4.19, we establish Theorem 2.9 for countably infinite partitions. Inparts of the existing literature (e.g., in Joyce 2009), credence functions are assumed to be definedon a (finite) partition of W to begin with, and so such a result might be especially relevant toextending the accuracy argument for probabilism to countably infinite opinion sets. Lemma 4.25.
A partition is W-stable relative to any generalized legitimate inaccuracy measure.
Proof.
See the Appendix.
Theorem 4.26.
Let ( W, F ) be a partition and I a generalized legitimate inaccuracy measure.Then the following are equivalent:1. c is coherent;2. c is not weakly dominated;3. c is not strongly dominated. Proof.
The result follows from Corollary 4.22, Lemma 4.25, and the fact that I ( c ∗ , · ) is boundedon W ∗ for each coherent credence function c . To see the latter, note that since c is coherent itfollows that P c i = P c ∗ i ≤ . For w ∈ W such that w ∈ p i , recalling that ϕ (0) = 0 , I ( c ∗ , w ) = a i d (1 , c ∗ i )+ X j = i a j d (0 , c ∗ j ) = a i d (1 , c ∗ i )+ X j = i a j ( c ∗ j ϕ ′ ( c ∗ j ) − ϕ ( c ∗ j )) ≤ C + D X j c ∗ j ≤ C + D for some constants C, D independent of c ∗ or w . Similarly, as seen in the proof of Lemma 4.25, W ∗ \ W = { w ∗ } where I ( c ∗ , w ∗ ) = ∞ X j =1 a j d (0 , c ∗ j ) = ∞ X j =1 a j ( c ∗ j ϕ ′ ( c ∗ j ) − ϕ ( c ∗ j )) ≤ C It has been noted that de Finetti’s (1974) original proof of Theorem 2.9 assuming the Brier score extends tocountably infinite opinion sets. However, the only proof we have seen is a sketch of the necessity of coherence forcountably infinite partitions by (Joyce, 1998, footnote 6), and such a claim could not be true for arbitrary countableopinion sets as Examples 4.1 and 4.2 show. Further, we prove the extension for arbitrary generalized legitimateinaccuracy measures. C independent of c ∗ or w . It follows that i) all coherent credence functions havefinite expected inaccuracy and ii) I ( c, w ) < ∞ for c ∈ C and w ∈ W . Thus Lemma 4.25 andCorollary 4.22 establish the result. In this section, we establish the equivalence between coherence and avoiding strong dominancefor certain opinion spaces (Theorem 4.29). The conditions are in terms of the analogous stabil-ity condition—
S-stability (Definition 4.28)—but a different finiteness assumption, and the proofstrategy is the same as for Theorem 4.19.We begin by establishing that on compact opinion spaces, coherent and thus countably coherentcredence functions (recall Theorem 4.11) are not strongly dominated.
Proposition 4.27.
Let ( W, F ) be a compact opinion space and I a generalized legitimate inac-curacy measure. If c is coherent (and thus countably coherent), then c is not strongly dominatedrelative to I . Proof.
See the Appendix.We now introduce S-stability and the main theorem of this section.
Definition 4.28.
Let ( W, F ) be S-stable relative to I if whenever a coherent credence function c defined on ( W, F ) is strongly dominated relative to I , then c ∗ defined on ( W ∗ , F ∗ ) is stronglydominated relative to I . Theorem 4.29.
Let I be a generalized legitimate inaccuracy measure and ( W, F ) an S-stableopinion space relative to I . Assume that I ( c, w ) < ∞ for each coherent credence function c and w ∈ W . Then the following are equivalent:1. c is coherent;2. c is not strongly dominated. Proof.
Assume c is coherent. c ∗ defined on the compact opinion space ( W ∗ , F ∗ ) is countablycoherent by Lemma 4.13. So by Proposition 4.27, c ∗ is not strongly dominated. But since F isS-stable this implies that c is not strongly dominated relative to W . The other direction followsfrom Theorem 3.4. Remark 4.30.
It is trivial to see that S-stability is necessary for the equivalence of coherence andavoiding strong dominance. It is open how much the assumption that coherent credence functionssatisfy I ( c, w ) < ∞ for all w can be weakened. Remark 4.31.
Schervish et al. (2009) take a different approach to dropping the assumption thatthe opinion set is finite: they apply weak and strong dominance notions to finite subsets of arbi-trarily sized opinion sets. They also explore connections between the two notions of dominanceconsidered here—weak and strong dominance—and what they call coherence , which amounts toavoiding being susceptible to a finite Dutch book . Thanks to Teddy Seidenfeld for pointing me to this work of Schervish et al.. An additional point worth notingabout their work is that they further generalize the finite results of Predd et al. (2009) by i) allowing a wider variety emark 4.32. Theorem 4.29 is related to Theorem 1 of Schervish et al. 2014. However, 1) theirassumptions are in some ways weaker and in some ways stronger than those in Theorem 4.29 and2) while Schervish et al. (2014) establish that coherence is sufficient for avoiding strong dominancein certain cases, unlike Theorems 4.19 and 4.29, their results do not show that coherence is sufficientfor avoiding even weak dominance in certain cases or that incoherence always entails being weaklydominated (and sometimes strongly dominated) by a coherent credence function (see Remark 3.6). While Theorems 4.19 and 4.29 come close to characterizing the opinion spaces on which not beingweakly and strongly dominated, respectively, are equivalent to coherence, it is open how far thefiniteness assumptions in the theorems can be weakened. This is a natural next line of inquiry.In addition, it would be useful to determine characterizations of W- and S-stability in terms ofthe inaccuracy measure that make it relatively easy to check whether an opinion set is W- orS-stable. Also, there are natural ways to generalize the results above to more closely match thefinite results: allow different one-dimensional Bregman divergences for different propositions andallow unbounded one-dimensional Bregman divergences.Another direction one could go in exploring the sufficiency of coherence for avoiding dominanceis as follows: instead of characterizing the countable opinion sets on which Theorem 2.9 goesthrough, one could characterize the kinds of coherent credence functions for which Theorem 2.9goes through on any countable opinion set. Doing so might show that while coherence is notenough to avoid dominance in all cases, coherence along with additional plausible constraints issufficient. In particular, while restricting to finitely supported credence functions is not enoughto establish the sufficiency of coherence for avoiding strong dominance (due to Example 4.2), itis open whether countable coherence is equivalent to avoiding weak or strong dominance on therestricted class.
So far we have been concerned with credences defined on countably infinite opinion sets. Wenow consider what can be said in favor of probabilism when credences are defined on uncountableopinion sets. When extending from the finite to the countably infinite setting, we used inaccuracymeasures that naturally restrict to legitimate inaccuracy measures in the finite case. Similarly, inthe uncountable case, we allow for measure theoretically defined inaccuracy measures that naturallyrestrict to generalized legitimate inaccuracy measures in the countable case. However, for the sakeof generality, we allow inaccuracy to be defined by integration against any finite measure. of inaccuracy measures including those which are merely proper as opposed to strictly proper and ii) by scoringconditional probabilities. A natural direction for future work is to use these relaxations in the finite case to relaxassumptions made here. Similarly, Steeger (2019) considers the property of avoiding strong dominance with respectto the Brier score for every finite subset of arbitrarily sized opinion sets (see “sufficient coherence” on p. 38). Schervish et al. require that the prevision for the inaccuracy of the credence function be finite and that inac-curacy be pointwise finite, while we only assume the latter. On the other hand, we require the opinion set to be S -stable while they do not. Thanks to Thomas Icard and Milan Mosse for suggesting this alternative direction of study. Note that since the counting measure over N is not a finite measure, the result below does not directly establishTheorem 3.4. Definition 5.1.
Let ( F , A , µ ) be a measure space and c : F → R + . If c is A -measurable and µ ( { p : c ( p ) / ∈ [0 , } ) = 0 , we call c a µ -credence function . We say a µ -credence function c is µ -coherent if there is a coherent (in the usual sense) credence function c ′ on F with c = c ′ µ -a.e.We say a µ -credence function is µ -incoherent if there is no coherent credence function c ′ such that c = c ′ µ -a.e. Definition 5.2.
Let F be an opinion set (of arbitrary cardinality) over a set W of worlds. Let ( F , A , µ ) be a σ -finite measure space over the opinion set F . Let C be the space of all µ -credencefunctions. Assume I : C × W → [0 , ∞ ] is such that, for all ( c, w ) ∈ C × W , we have I ( c, w ) = B ϕ,µ ( v w , c ) , where B ϕ,µ is a Bregman distance relative to ϕ and ( F , A , µ ) (see Definition A.1). In particular,each v w is a µ -credence function. Then we call I an integral inaccuracy measure on ( F , A , µ ) .We now state a dominance result about integral inaccuracy measures. The proof is essentiallya measure theoretic version of the proof of Theorem 3.4. Theorem 5.3.
Let I be an integral inaccuracy measure on a finite measure space ( F , A , µ ) .Then for every µ -credence function c , if c is µ -incoherent, then there is a µ -coherent µ -credencefunction c ′ that strongly dominates c relative to I . Proof.
See the Appendix.Here is an example of how Theorem 5.3 can be used to give an accuracy argument in a concreteuncountable setting. Assume we have a coin with unknown bias θ ∈ [0 , and a set of propositionsof the form “ a ≤ θ ≤ b ” for each a, b ∈ [0 , with a ≤ b . Then a credence function on thisuncountable opinion set can be represented by a function c : X → [0 , , where X = { ( a, b ) : 0 ≤ a ≤ b ≤ } ⊆ [0 , . We put the Lebesgue measure λ on X to generalizethe additive constraint often assumed in the finite case. We let I ( c, w ) = Z X d ( v w ( x ) , c ( x )) λ ( d x ) for a bounded one-dimensional Bregman divergence d . Then the assumptions of Theorem 5.3 hold,so we get the following dominance result: for any λ -credence function c , if c is a λ -incoherent, thenthere is a λ -coherent λ -credence function that strongly dominates c . Again, we assume the one-dimensional Bregman divergence d generated by ϕ is bounded. Conclusion
There is plenty of normative work to be done using the results established above. In light ofthe failure of coherence being sufficient to avoid strong dominance on certain countably infiniteopinion sets, the most pressing question seems to be: is there an accuracy-based argument forprobabilism on at least all countable opinion sets? If not, what does this mean for the accuracyproject as a whole? Can we give some sort of privileged status to certain kinds of opinion sets orinaccuracy measures for which coherence is equivalent to not being dominated, e.g., partitions?What is the normative status of the stronger condition of countable coherence? Further, whilethe measure theoretic framework introduced in Section 5 to score inaccuracy of credence functionsover opinion sets of arbitrary cardinality seems like a natural extension of the finite and countablyinfinite frameworks, is it well motivated that inaccuracy does not track the behavior of a credencefunction on measure zero sets? The hope with this paper is to start a conversation about thesequestions by first establishing relevant mathematical results.
Acknowledgements
Thanks to participants of the Berkeley-Stanford Logic Circle (April 2019), Probability and LogicConference (July 2019), Berkeley Formal Epistemology Reading Course (October 2019), and Stan-ford Logic and Formal Philosophy Seminar (November 2019), to whom earlier versions of this pa-per were presented. Special thanks to Craig Evans, Wesley Holliday, Thomas Icard, Kiran Luecke,Calum McNamara, Sven Neth, Richard Pettigrew, Eric Raidl, Teddy Seidenfeld, and James Walshfor helpful comments and discussion.
A Appendix
A.1 Proof of Theorem 3.4
We review the necessary background before proving Theorem 3.4.
A.1.1 Generalized Projections
Csiszár (1995) showed that what he calls generalized projections onto convex sets with respect toBregman distances exist under very general conditions. We review his relevant results here (butassume knowledge of basic measure theory).
Definition A.1.
Fix a σ -finite measure space ( X, X , µ ) . The Bregman distance of non-negative( X -measurable) functions s and t is defined by B ϕ,µ ( s, t ) = Z d ( s ( x ) , t ( x )) µ ( dx ) ∈ [0 , ∞ ] where d ( s ( x ) , t ( x )) = ϕ ( s ( x )) − ϕ ( t ( x )) − ϕ ′ ( t ( x ))( s ( x ) − t ( x )) for some strictly convex, differentiablefunction ϕ on (0 , ∞ ) . Note that B ϕ,µ ( s, t ) = 0 iff s = t µ -a.e. See Csiszár 1995, p. 165 for details. For B ϕ,µ to be a distance measure, we do not need to assume that ϕ (1) = ϕ ′ (1) = 0 by the remark following(1.9) in Csiszár 1995. emark A.2. Notice that a generalized quasi-additive Bregman divergence D with weights { a i } ∞ i =1 whose generating one-dimensional Bregman divergence d is given in terms of ϕ has acorresponding Bregman distance B ¯ ϕ,µ with1. the measure space being ( N , P ( N ) , µ ) , where µ ( A ) = P i ∈ A a i for each A ∈ P ( N ) , and2. ¯ ϕ on (0 , ∞ ) being a strictly convex, differentiable extension of ϕ on [0 , . Thus non-negative ( P ( N ) -measurable) functions are elements of R + ∞ . Note, importantly, thatthe corresponding generalized legitimate inaccuracy measure I determined by D is also given bythe corresponding Bregman distance. That is, I ( c, w ) = B ¯ ϕ,µ ( v w , c ) . To simplify notation, let B denote B ¯ ϕ,µ a Bregman distance. Let S be the set of non-negativemeasurable functions. For any E ⊆ S and t ∈ S , we write B ( E, t ) = inf s ∈ E B ( s, t ) . If there exists s ∗ ∈ E with B ( s ∗ , t ) = B ( E, t ) , then s ∗ is unique and is called the B-projection of t onto E (see Csiszár 1995, Lemma 2). As Csiszár notes, these projections may not exist. However, aweaker kind of projection exists in a large number of cases. To describe them, we need to introducea kind of convergence called loose in µ -measure convergence . Definition A.3.
We say a sequence { s n } of elements from S converges loosely in µ -measure to t ,denoted by s n µ t , if for every A ∈ X with µ ( A ) < ∞ , we have lim n →∞ µ ( A ∩ { p : | s n ( p ) − t ( p ) | > ǫ } ) = 0 for all ǫ > . Definition A.4. i. Given E ⊆ S and t ∈ S , we say that a sequence { s n } of elements from E is a B -minimizing sequence if B ( s n , t ) → B ( E, t ) .ii. If there is an s ∗ ∈ S such that every B -minimizing sequence converges to s ∗ loosely in µ -measure, then we call s ∗ the generalized B -projection of t onto E .The result that is integral to proving Theorem 3.4 is the following (see Csiszár’s Theorem 1,Lemma 2, and Corollary of Theorem 1). Theorem A.5 (Csiszár 1995) . Let E be a convex subset of S and t ∈ S . If B ( E, t ) is finite, thenthere exists s ∗ ∈ S such that B ( s, t ) ≥ B ( E, t ) + B ( s, s ∗ ) for every s ∈ E and B ( E, t ) ≥ B ( s ∗ , t ) . It follows that the generalized B -projection of t onto E exists and equals s ∗ . Using that ϕ ′ exists and is finite at x = 1 as we assumed d is bounded, we extend ϕ as follows: for x ∈ [1 , ∞ ) ,let ¯ ϕ ( x ) = q ( x ) = x + bx + c , where b and c are chosen so ϕ (1) = q (1) and ϕ ′ (1) = q ′ (1) . Then using the factthat ¯ ϕ is differentiable at by construction and a function is strictly convex if and only if its derivative is strictlyincreasing, it is easy to see that ¯ ϕ is differentiable and strictly convex on (0 , ∞ ) . .1.2 Extending Partial Measures We also use an extension result of Horn and Tarski (1948) in the proof of Theorem 3.4. FollowingHorn and Tarski, we introduce partial measures and recall that they can be extended to finitelyadditive probability functions. Recall the definition of a finitely additive probability function inDefinition 2.2 (though we drop the assumption that F is finite). Remark A.6.
It is a simple corollary of the definition of a finitely additive probability function c over an algebra F that for any p, p ′ ∈ F : if p ⊆ p ′ , then c ( p ) ≤ c ( p ′ ) .Here is another useful fact about finitely additive probability functions. Proposition A.7. If c is a finitely additive probability function on an algebra F and a , . . . , a m − ∈F , then m − X k =0 c ( a k ) = m − X k =0 c ( [ p ∈ S m,k \ i ≤ k a p i ) (4)where S m,k is the set of all sequences p = ( p , . . . , p k ) with ≤ p < . . . < p k < m .To introduce the notion of a partial measure, we need the following definition. Definition A.8.
Let ϕ , . . . , ϕ m − and ψ , . . . , ψ n − be elements of F . Then we write ( ϕ , . . . , ϕ m − ) ⊆ ( ψ , . . . , ψ n − ) to mean [ p ∈ S m,k \ i ≤ k ϕ p i ⊆ [ p ∈ S n,k \ i ≤ k ψ p i for every k < m (5)where S r,k ( r = m, n ) is as in Proposition A.7. Definition A.9.
A function c , defined on a subset S of an algebra F over W , that maps to R iscalled a partial measure if it satisfies the following properties:1. c ( x ) ≥ for x ∈ S ;2. If ϕ , . . . , ϕ m − , ψ , . . . , ψ n − ∈ S and ( ϕ , . . . , ϕ m − ) ⊆ ( ψ , . . . , ψ n − ) , then m − X k =0 c ( ϕ k ) ≤ n − X k =0 c ( ψ k ); W ∈ S and c ( W ) = 1 .The following result is the point of introducing the above definitions. Theorem A.10 (Horn and Tarski 1948) . Let c be a partial measure on a subset F of an algebra A . Then there is a finitely additive probability function c ∗ on A that extends c . Note that if m > n , this condition implies S p ∈ S m,k T i ≤ k ϕ p i = S p ∈ S n,k T i ≤ k ψ p i = ∅ for k ≥ n . .1.3 Proof We now establish the necessity of coherence to avoid dominance.
Theorem 3.4.
Let F be a countably infinite opinion set, I a generalized legitimate inaccuracymeasure, and c an incoherent credence function. Then:1. c is weakly dominated relative to I by a coherent credence function; and2. if I ( c, w ) < ∞ for each w ∈ W , then c is strongly dominated relative to I by a coherentcredence function. Proof.
Let I be a generalized legitimate inaccuracy measure and thus defined by a Bregmandistance B ¯ ϕ,µ (see Remark A.2). We write B for B ¯ ϕ,µ . Let S be the set of non-negative functionson F . Let E ⊆ S be the set of coherent credence functions on F . Then clearly E is convex.Let c be an incoherent credence function. Case 1 : I ( c, w ) = ∞ for all w ∈ W . Then since I ( v w , w ) = 0 for all w ∈ W , any omniscientcredence function weakly dominates c . Case 2 : I ( c, w ′ ) < ∞ for some w ′ ∈ W . We show that there is a coherent credence function π c such that I ( c, w ) > I ( π c , w ) for any w such that I ( c, w ) < ∞ . Since v w ′ ∈ E , we see that B ( E, c ) ≤ B ( v w ′ , c ) = I ( c, w ′ ) < ∞ . Thus we can apply Theorem A.5 to get a π c ∈ S such that B ( s, t ) ≥ B ( E, c ) + B ( s, π c ) for every s ∈ E. (6)In particular, (6) holds when s is the omniscient credence function at world w for any w ∈ W ; andso we see that I ( c, w ) ≥ B ( E, c ) + I ( π c , w ) (7)for all w , where all numbers in (7) are finite whenever I ( c, w ) < ∞ .Next we show that π c is in fact coherent. This is due to the following claim: E is closed underloose convergence in µ -measure where µ is a weighted counting measure on P ( N ) defined withweights { a i } ∞ i =1 . To see this, let c n ∈ E for each n and c ∈ S . Assume c n → c loosely in µ -measure.We show c ∈ E , i.e., c is coherent. Note c is coherent on F if and only if c ′ : F ∪ { W } → [0 , iscoherent on F ∪ { W } , where c ′ = c on F and c ′ ( W ) = 1 . Thus it suffices to assume c and c n forall n are defined on F ∪ { W } with c ( W ) = c n ( W ) = 1 for all n .It is easy to see that loose convergence in a weighted counting measure (where all weights arenon-zero) implies pointwise convergence on F , so c ( p ) = lim n →∞ c n ( p ) ∈ [0 , for each p ∈ F ∪ { W } . To show c ∈ E , it suffices to show c can be extended to a finitely additiveprobability function on P ( W ) . 20e first show c is a partial measure on F ∪ { W } . Definitions A.9.1 and A.9.3 clearly hold for c so we just need to show Definition A.9.2 holds. Let ϕ , . . . , ϕ m − , ψ , . . . , ψ m ′ − ∈ F ∪ { W } and [ p ∈ S m,k \ i ≤ k ϕ p i ⊆ [ p ∈ S m ′ ,k \ i ≤ k ψ p i for every k < m . Since the c n are coherent and thus extend to finitely additive probability functionson algebras containing F , we have by Proposition A.7 and Remark A.6 that m − X k =0 c n ( ϕ k ) = m − X k =0 c n ( [ p ∈ S m,k \ i ≤ k ϕ p i ) ≤ m ′ − X k =0 c n ( [ p ∈ S m ′ ,k \ i ≤ k ψ p i ) = m ′ − X k =0 c n ( ϕ k ) using that [ p ∈ S m,k \ i ≤ k ψ p i = [ p ∈ S m ′ ,k \ i ≤ k ψ p i = ∅ for k ≥ m ′ . Sending n to infinity and using the pointwise convergence of c n to c on F ∪ { W } weobtain that m − X k =0 c ( ϕ k ) ≤ m ′ − X k =0 c ( ψ k ) . Thus c is a partial measure on F ∪{ W } . By Theorem A.10, it follows that there is a finitely additiveprobability function c ∗ on an algebra F ∗ ⊇ F that extends c and so c ∈ E , which concludes theproof that E is closed under loose µ -convergence.By Theorem A.5, π c is the generalized B -projection of c onto E . Also, since B ( E, c ) = inf s ∈ E ( s, c ) < ∞ , there is a B-minimizing sequence { s n } ⊆ E such that B ( s n , c ) → B ( E, c ) by the definition ofinfimum. By the definition of a generalized projection, s n µ π c . Since E is closed under looseconvergence, it follows that π c ∈ E . Further, by Theorem A.5, B ( E, c ) ≥ B ( π c , c ) > , since π c = c (as c is incoherent) and B ( s, t ) = 0 if and only if s = t (as µ is a weighted countingmeasure with all non-zero weights). So for every w such that I ( c, w ) < ∞ , we deduce that I ( c, w ) ≥ B ( E, c ) + I ( π c , w ) > I ( π c , w ) . This proves that c is weakly dominated by π c , and c is strongly dominated by π c if I ( c, w ) < ∞ for all w ∈ W . 21 .2 Proofs from Section 4 Proposition 4.7.
Let F be a countably discriminating opinion set. Then a credence function c is countably coherent if and only if there are λ w ∈ [0 , with P v w ∈V F λ w = 1 such that c ( p ) = X v w ∈V F λ w v w ( p ) for all p ∈ F . Proof.
We adapt the proof of Proposition 1 in Predd et al. 2009. Let F = { p , p , . . . } . Let X be the collection of all nonempty sets of the form T ∞ i =1 p ∗ i where p ∗ i is either p i or p ci . Then X partitions W . Also, X is in bijection with V F , the set of omniscient credence functions.Indeed, let f map v w to T ∞ i =1 p ∗ i where p ∗ i = p i if v w ( p i ) = 1 and p ∗ i = p ci otherwise. Then foreach w , w ∈ f ( v w ) and so f ( v w ) ∈ X . Note f is onto. Indeed, let w ∈ T ∞ i =1 p ∗ i , where T ∞ i =1 p ∗ i ∈ X .Then f ( v w ) = T ∞ i =1 p ∗ i . Also, f is injective. Indeed, assume f ( v w ) = f ( v w ′ ) . Then f ( v w ) = ∞ \ i =1 p i = ∞ \ i =1 p i = f ( v w ′ ) for p ji = p i or p ji = p ci for all i ∈ N and j ∈ { , } . If p i = p i for some i , then without loss ofgenerality we may assume p i = p i and p i = p ci . So w ∈ p i but w / ∈ p i and thus w / ∈ T ∞ i =1 p i . But w ∈ T ∞ i =1 p i by definition of f and so T ∞ i =1 p i = T ∞ i =1 p i , which is a contradiction. It follows that p i = p i for all i , but then by definition of f , this implies v w ( p i ) = 1 if and only if v w ′ ( p i ) = 1 forall i and so v w = v w ′ .It is easy to see that since F is countably discriminating, V F is countable. It follows that X iscountable. Enumerate the elements of V F and X by v w , v w , . . . and e , e , . . . , respectively, suchthat f − ( e j ) = v w j . We have that p i is the disjoint union of e j such that e j ⊆ p i , or equivalentlythe e j where f − ( e j )( p i ) = 1 . Note i) for any countably additive probability function µ on a σ -algebra containing F (and thus containing X ) and any p i ∈ F : µ ( p i ) = ∞ X j =1 µ ( e j ) f − ( e j )( p i ) . Now we prove the equivalence. Assume c is countably coherent. So c extends to a countablyadditive probability function µ on a σ -algebra containing F . Then by i), c ( p i ) = µ ( p i ) = ∞ X j =1 µ ( e j ) f − ( e j )( p i ) for all p i ∈ F . But since µ ( e j ) are non-negative and sum to (since the e j ’s partition W and µ isa countably additive probability function), we have that c has the form stated.Now assume c ( p i ) = P ∞ j =1 λ j v w j ( p i ) for all i where P ∞ j =1 λ j = 1 . Let σ ( F ) be the smallest σ -algebra on W containing F . Then it is easy to check that the function on σ ( F ) defined by ¯ v w j ( p ) = 1 if and only if w j ∈ p extends v w j and is a countably additive probability function on σ ( F ) . Then P ∞ j =1 λ j ¯ v w j is a countably additive probability function on σ ( F ) since a countablesum of countably additive probability functions with coefficients that sum to is a countably22dditive probability function. Since c ( p i ) = ∞ X j =1 λ j v w j ( p i ) = ∞ X j =1 λ i ¯ v w j ( p i ) for all i , it follows that c extends to a countably additive probability function on a σ -algebracontaining F . Lemma 4.12.
For any opinion space ( W, F ) , ( W ∗ , F ∗ ) is compact. Proof.
Let { Ψ( p n ) f ( n ) } ∞ n =1 be a sequence of elements of F ∗ or their complements as in Definition4.9. Case 1: for each N there is some w N ∈ W such that w N ∈ T Nn =1 Ψ( p n ) f ( n ) . Then since i) Ψ( p ) ∩ W = p and ii) Ψ( p ) c ∩ W = p c for any p ∈ F , it follows that w N ∈ T Nn =1 p f ( n ) n for each N . Ifthere is some w ′ ∈ W with w ′ ∈ T ∞ n =1 p f ( n ) n then by i) and ii) it follows that w ′ ∈ T ∞ n =1 Ψ( p n ) f ( n ) .Otherwise, by construction, we defined some x s to be such that x s ∈ T ∞ n =1 Ψ( p n ) f ( n ) . In eithercase, we are done. Case 2: there is some N such that T Nn =1 Ψ( p n ) f ( n ) ⊆ W ∗ \ W . I claim this impliesthat T Nn =1 Ψ( p n ) f ( n ) = ∅ . Indeed, if there were some w ∈ W ∗ \ W such that w ∈ T Nn =1 Ψ( p n ) f ( n ) ,then that is because { p f ( n ) n } Nn =1 is an initial sequence of some sequence { ¯ p ¯ f ( n ) n } ∞ n =1 such that T ln =1 ¯ p ¯ f ( n ) n = ∅ for each l and thus, in particular, T Nn =1 p f ( n ) n = ∅ . So there is some w ∈ W suchthat w ∈ T Nn =1 Ψ( p n ) f ( n ) by i) and ii), which is a contradiction. Thus we have established that ( W ∗ , F ∗ ) is compact. Lemma 4.13.
Let ( W, F ) be an opinion space and c a coherent credence function on ( W, F ) . Let ( W ∗ , F ∗ ) be the compactification of ( W, F ) and define c ∗ (Ψ( p )) := c ( p ) for each p ∈ F . Then c ∗ is a countably coherent credence function on ( W ∗ , F ∗ ) and I ( c, w ) = I ( c ∗ , w ) for w ∈ W . Proof.
Since ( W ∗ , F ∗ ) is compact, we only need to show that c ∗ is coherent by Theorem 4.11. Thusit suffices to show that c ∗ can be extended to a finitely additive probability function on A ( F ∗ ) .Since c is coherent, there is a finitely additive probability function ¯ c such that:1. ¯ c ( p ) = c ( p ) for p ∈ F ;2. ¯ c ( p ∪ q ) = ¯ c ( p ) + ¯ c ( q ) for p, q ∈ F with p ∩ q = ∅ ;3. ¯ c ( W ) = 1 .First, define Ψ( p c ) := Ψ( p ) c for each p ∈ F . Then each element in A ( F ∗ ) can be represented by S Ni =1 T Mj =1 Ψ( q ij ) where q ij or its complement is in F . We define ¯ c ∗ ( N [ i =1 M \ j =1 Ψ( q ij )) := ¯ c ( N [ i =1 M \ j =1 q ij ) . Using that p = Ψ( p ) ∩ W and p c = Ψ( p ) c ∩ W , we show that ¯ c ∗ is a well-defined finitely additiveprobability function on A ( F ∗ ) extending c ∗ . We first show ¯ c ∗ is well-defined. Assume that N [ i =1 M \ j =1 Ψ( q ij ) = N ′ [ i =1 M ′ \ j =1 Ψ( r ij ) . N [ i =1 M \ j =1 Ψ( q ij ) ∩ W = N ′ [ i =1 M ′ \ j =1 Ψ( r ij ) ∩ W which, noting that p = Ψ( p ) ∩ W and p c = Ψ( p ) c ∩ W , establishes that N [ i =1 M \ j =1 q ij = N ′ [ i =1 M ′ \ j =1 r ij , and so ¯ c ∗ ( N [ i =1 M \ j =1 Ψ( q ij )) = ¯ c ( N [ i =1 M \ j =1 q ij ) = ¯ c ( N ′ [ i =1 M ′ \ j =1 r ij ) = ¯ c ∗ ( N ′ [ i =1 M ′ \ j =1 Ψ( r ij )) . Thus ¯ c ∗ is well-defined. Clearly, ¯ c ∗ extends c ∗ . Now, since W ⊆ W ∗ , if N [ i =1 M \ j =1 Ψ( q ij ) ∩ N ′ [ i =1 M ′ \ j =1 Ψ( r ij ) = ∅ then N [ i =1 M \ j =1 Ψ( q ij ) ∩ W ∩ N ′ [ i =1 M ′ \ j =1 Ψ( r ij ) ∩ W = ∅ and so ¯ c ( N [ i =1 M \ j =1 q ij ∪ N ′ [ i =1 M ′ \ j =1 r ij ) = ¯ c ( N [ i =1 M \ j =1 q ij ) + ¯ c ( N ′ [ i =1 M ′ \ j =1 r ij ) . Then noting the definition of ¯ c ∗ in terms of ¯ c , we establish finite additivity. Lastly, if W ∗ = N [ i =1 M \ j =1 Ψ( q ij ) , then W = N [ i =1 M \ j =1 Ψ( q ij ) ∩ W, and so ¯ c ∗ ( N [ i =1 M \ j =1 Ψ( q ij )) = ¯ c ( N [ i =1 M \ j =1 q ij ) = ¯ c ( W ) = 1 . This establishes that c ∗ is coherent on ( W ∗ , F ∗ ) , and so since ( W ∗ , F ∗ ) is compact, c ∗ is countablycoherent. Further, w ∈ p if and only if w ∈ Ψ( p ) for each w ∈ W , so v w defined on F is the sameas v w defined on F ∗ for each w ∈ W . Since c ( p ) = c ∗ (Ψ( p )) for all p ∈ F , this establishes that I ( c, w ) = I ( c ∗ , w ) for each w ∈ W . Proposition 4.16.
Let ( W, F ) be an opinion space with F countably infinite and I a generalizedlegitimate inaccuracy measure. If c is a countably coherent credence function with finite expectedinaccuracy, then c is not weakly dominated. 24 roof. Since c is countably coherent, let ¯ c be a countably additive probability function on σ ( F ) extending c such that E ¯ c I ( c, · ) < ∞ . Note that since d is strictly proper (see Remark 2.7), weknow that for any i ∈ N , E ¯ c d ( v w , c i ) < E ¯ c d ( v w , x ) for x = c i . Assume toward a contradiction thatthere is a credence function d with d = c and I ( d, w ) ≤ I ( c, w ) for each w with strict inequalityfor some w . Then E ¯ c I ( d, · ) ≤ E ¯ c I ( c, · ) < ∞ , so both I ( d, · ) and I ( c, · ) are integrable withrespect to the measure space ( W, σ ( F ) , ¯ c ) . Then let i be any index such that d i = c i . There mustbe at least one since c = d . Then E ¯ c d ( v w , c i ) < E ¯ c d ( v w , d i ) . If d i = c i then clearly E ¯ c d ( v w , c i ) = E ¯ c d ( v w , d i ) . So since E ¯ c I ( c, · ) < ∞ and E ¯ c I ( d, · ) < ∞ , wehave E ¯ c I ( c, · ) = ∞ X i =1 a i E ¯ c d ( v w , c i ) < ∞ X i =1 a i E ¯ c d ( v w , d i ) = E ¯ c I ( d, · ) , which implies that E ¯ c ( I ( c, · ) − I ( d, · )) < . Thus there is some nonempty set E ∈ σ ( F ) with ¯ c ( E ) > on which I ( c, · ) − I ( d, · ) < (since the Lebesgue integral is positive). But thiscontradicts our assumption that d weakly dominates c , and so we are done. Proposition 4.17.
Let ( W, F ) be a point-finite opinion space with F countably infinite and I a generalized legitimate inaccuracy measure. If a credence function c is countably coherent andsomewhere finitely inaccurate relative to B , then c is not weakly dominated relative to B . Proof.
Assume d weakly dominates c . Note i) c is somewhere finitely inaccurate if and only if B ( c, w ) < ∞ for all w ∈ W if and only if P ∞ i =1 c i < ∞ . It follows by weak dominance that B ( d, w ) < ∞ for all w ∈ W and therefore P ∞ i =1 d i < ∞ . Let B ( c, w ) = D ( v w , c ) for D ageneralized quasi-additive Bregman divergence.Since ( W, F ) is point-finite, it is also countably discriminating as there are only countably manyfinite subsets of F . So by Proposition 4.7, c = P ∞ j =1 λ j v w j for λ j ∈ [0 , with P ∞ j =1 λ j = 1 . First,note that D ( P ∞ j =1 λ j v w j , c ) = 0 and ( P ∞ j =1 λ j v w j ( p i ) − c ( p i )) = 0 for all i , so D ( ∞ X j =1 λ j v w j , c ) − D ( ∞ X j =1 λ j v w j , d ) = ∞ X i =1 a i ( ∞ X j =1 λ j v w j ( p i ) − c i ) − ( ∞ X j =1 λ j v w j ( p i ) − d i ) . Using that ( ∞ X j =1 λ j v w j ( p i ) − c i ) − ( ∞ X j =1 λ j v w j ( p i ) − d i ) = ∞ X j =1 λ j [( v w j ( p i ) − c i ) − ( v w j ( p i ) − d i ) ] i since P ∞ j =1 λ j = 1 , we have that D ( ∞ X j =1 λ j v w j , c ) − D ( ∞ X j =1 λ j v w j , d ) = ∞ X i =1 a i ∞ X j =1 λ j [( v w j ( p i ) − c i ) − ( v w j ( p i ) − d i ) ] (8) = ∞ X i =1 a i ( X j : w j / ∈ p i λ j )( c i − d i ) + a i ( X j : w j ∈ p i λ j )((1 − c i ) − (1 − d i ) )= ∞ X i =1 a i ( c i − d i ) + 2 a i ( X j : w j ∈ p i λ j )( d i − c i )= ∞ X i =1 a i ( − c i − d i ) + 2 a i ( X j : w j ∈ p i λ j ) d i since c i = P j : w j ∈ p i λ j . We have P ∞ i =1 c i + d i < ∞ by i). Thus ≤ ∞ X i =1 a i X j : w j ∈ p i λ j ) d i < ∞ (9)because ≥ D ( ∞ X j =1 λ j v w j , c ) − D ( ∞ X j =1 λ j v w j , d ) . Having established (9), we claim we can use the dominated convergence theorem (see, e.g.,Theorem 1.4.49 in Tao 2011) to switch limits in (8). Indeed, ∞ X i =1 a i N X j =1 λ j [( v w j ( p i ) − c i ) − ( v w j ( p i ) − d i ) ] = ∞ X i =1 a i ( X ≤ j ≤ N λ j )( c i − d i ) + 2 a i ( X j : w j ∈ p i ≤ j ≤ N λ j )( d i − c i ) . Letting g N ( i ) = a i ( X ≤ j ≤ N λ j )( c i − d i ) + 2 a i ( X j : w j ∈ p i ≤ j ≤ N λ j )( d i − c i ) and noting that − ( P j : w j ∈ p i ≤ j ≤ N λ j ) c i ≥ − c i since c i = P j : w j ∈ p i λ j , we see that | g N ( i ) | ≤ a i (2 c i + d i + 2 a i ( X j : w j ∈ p i λ j ) d i ) . Each of c i , d i , and ( P j : w j ∈ p i λ j ) d i is summable in i and sup i a i < ∞ . So, the dominated conver-gence theorem applies, and we can switch limits.26hus we have ≥ D ( ∞ X j =1 λ j v w j , c ) − D ( ∞ X j =1 λ j v w j , d )= ∞ X j =1 λ j ∞ X i =1 a i [( v w j ( p i ) − c ( p i )) − ( v w j ( p i ) − d ( p i )) ]= ∞ X j =1 λ j ( D ( v w , c ) − D ( v w , d )) ≥ where we used that c and d are both finitely inaccurate for each w ∈ W to break up the summationin the second line. Thus we conclude that c = d , as D ( c, d ) = 0 if and only if c = d . Lemma 4.25.
A partition is W-stable relative to any generalized legitimate inaccuracy measure.
Proof.
Let F = { p , p , . . . } be a partition. Assume a coherent credence function c on F is weaklydominated by some credence function d . We can assume d is coherent by Theorem 3.4, and so P ∞ m =1 d m ≤ . We show that c ∗ is weakly dominated by d ∗ , thereby establishing that a partitionis W-stable.First, I ( c ∗ , w ) = I ( c, w ) and I ( d ∗ , w ) = I ( d, w ) for all w ∈ W by Lemma 4.13. Thus byassumption of weak dominance, I ( c ∗ , w ) ≥ I ( d ∗ , w ) for all w ∈ W with a strict inequality for some w ∈ W . We therefore need to only check what happens for w ∈ W ∗ \ W . The compactification of a partition consists in adding one point w ∗ which is in thecomplement of all p ∗ ∈ F ∗ . So W ∗ = W ∪ w ∗ and I ( c ∗ , w ∗ ) − I ( d ∗ , w ∗ ) = ∞ X m =1 a m ( ϕ ( d m ) − ϕ ′ ( d m ) d m ) − ∞ X m =1 a m ( ϕ ( c m ) − ϕ ′ ( c m ) c m ) , which I claim is greater than or equal to . Indeed, assume toward a contradiction that ∞ X m =1 a m ( ϕ ( d m ) − ϕ ′ ( d m ) d m ) < ∞ X m =1 a m ( ϕ ( c m ) − ϕ ′ ( c m ) c m ) . Then since d n → as P ∞ n =1 d n ≤ and c n → as P ∞ n =1 c n ≤ , ϕ ′ ( d n ) − ϕ ′ ( c n ) → (since ϕ ′ (0) = lim x → ϕ ( x ) = 0 ) and so we can find a K such that | ϕ ′ ( d n ) − ϕ ′ ( c n ) | < | ∞ X m =1 a m ( ϕ ( d m ) − ϕ ′ ( d m ) d m ) − ∞ X m =1 a m ( ϕ ( c m ) − ϕ ′ ( c m ) c m ) | for n ≥ K . Thus for n ≥ K , I ( c, w n ) − I ( d, w n ) = ∞ X m =1 a m ( ϕ ( d m ) − ϕ ′ ( d m ) d m ) − ∞ X m =1 a m ( ϕ ( c m ) − ϕ ′ ( c m ) c m )+ ϕ ′ ( d n ) − ϕ ′ ( c n ) < , contradicting that d weakly dominates c . So indeed, d ∗ weakly dominates c ∗ .27 roposition 4.27. Let ( W, F ) be a compact opinion space and I a generalized legitimate inac-curacy measure. If c is coherent (and thus countably coherent), then c is not strongly dominatedrelative to I . Proof.
Let I n ( c ′ , w ) := P ni =1 a i d ( v w ( p i ) , c ′ ( p i )) for each n ∈ N , w ∈ W , and credence function c ′ on F . Consider a credence function d = c . Define T n = { ( v w ( p ) , . . . , v w ( p n )) : I k ( c, w ) < I k ( d, w ) for some k ≥ n, w ∈ W } and T = { e } ∪ T ∞ n =1 T n , where e is the empty sequence. For each s, t ∈ T , we set s < t if and onlyif s is an initial sequence of t , and we set the height of t ∈ T to be the length of the tuple. Then T is a binary tree.We claim T is infinite. Fix n ∈ N . Then there is a t ∈ T with height n if and only if T n = ∅ if and only if I k ( c, w ) < I k ( d, w ) for some k ≥ n and w ∈ W . Let k be the maximum of n andthe smallest i such that c ( p i ) = d ( p i ) . Then since c restricted to any subset of F is coherent, byTheorem 2.9, I k ( c, w ′ ) < I k ( d, w ′ ) for some w ′ ∈ W and so ( v w ′ ( p ) , . . . , v w ′ ( p n )) ∈ T n .By Konig’s lemma (see, e.g., Hrbacek and Jech 1999, Sec. 12.3), there exists an infinite branch B = ∞ [ n =1 { ( v w n ( p ) , . . . , v w n ( p n )) } through T , where ( v w n ( p ) , . . . , v w n ( p n )) < ( v w m ( p ) , . . . , v w m ( p m )) whenever n < m . For each i , let p ∗ i = p i if v w i ( p i ) = 1 and p ∗ i = p ci if v w i ( p i ) = 0 . Then w n ∈ T ni =1 p ∗ i since v w i ( p i ) = 1 if and only if v w n ( p i ) = 1 for i < n as ( v w i ( p ) , . . . , v w i ( p i )) < ( v w n ( p ) , . . . , v w n ( p n )) . Thus T ni =1 p ∗ i = ∅ for each n and so by compactness there is some w ∈ T ∞ i =1 p ∗ i . Then ( v w ( p ) , . . . , v w ( p n )) = ( v w n ( p ) , . . . , v w n ( p n )) ∈ T n for each n ∈ N . By the definition of T n , for each n ∈ N we have I k n ( c, w ) < I k n ( d, w ) for some k n ≥ n . Sending n to infinity, I ( c, w ) ≤ I ( d, w ) and thus d does not strongly dominate c . A.3 Proof of Theorem 5.3
Theorem 5.3.
Let I be an integral inaccuracy measure on a finite measure space ( F , A , µ ) .Then for every µ -credence function c , if c is µ -incoherent, then there is a µ -coherent µ -credencefunction c ′ that strongly dominates c relative to I . Proof.
Let I ( c, w ) = B ϕ,µ ( v w , c ) . We write B for B ϕ,µ . Let S be the set of non-negative A -measurable functions on F . Let E ⊆ S be the set of µ -coherent µ -credence functions over F .Then E is convex. Let c be a µ -incoherent µ -credence function. Because µ is finite and d is28ounded, B ( E, c ) < ∞ . Thus we can apply Theorem A.5 to get a π c ∈ S such that B ( s, c ) ≥ B ( E, c ) + B ( s, π c ) for every s ∈ E. (10)In particular, (10) holds when s is the omniscient credence function at world w for each w , so weobtain I ( c, w ) ≥ B ( E, c ) + I ( π c , w ) (11)for all w , where all numbers in (11) are finite. We show that π c is in fact a µ -coherent µ -credencefunction. It suffices to show that π c is µ -a.e. equal to a coherent credence function on F (since π c ∈ S , it is A -measurable). To do so, we prove the following claim: E is closed under loose-convergence in µ -measure.To see this, let c n ∈ E for each n and c ∈ S . Assume c n → c loosely in µ -measure. Thefirst thing to notice is that, since µ is finite, loose µ -convergence implies µ -a.e. convergence on asubsequence { a n } ∞ n =1 of { n } ∞ n =1 , so that c ( p ) = lim n →∞ c a n ( p ) ∈ [0 , for each p ∈ G with µ ( G c ) = 0 . Since the c a n are µ -coherent, we can change each c a n on a(measurable) measure zero set X n to get coherent µ -credence functions c a n . Further, we replace G with G \ ( ∪ ∞ n =1 X n ) . Assuming these adjustments have been made, we have that c a n → c on G with µ ( G c ) = 0 , and each c a n is coherent. We now show c ∈ E by showing it is equal to a coherentcredence function on F when restricting to G .First, we extend c (resp. c a n ) to c (resp. c a n ), where c (resp. c a n ) is a credence function on G ∪ { W } such that c = c (resp. c a n = c a n ) on G and c ( W ) = 1 (resp. c a n ( W ) = 1 ). Then noticethat c (resp. c a n ) is coherent on G if and only if c (resp. c a n ) is coherent on G ∪ { W } . Thus wework with c and c a n instead noting that c = lim n c a n on G ∪ { W } . To show c ∈ E , we first show c is a partial measure on G ∪ { W } .Definitions A.9.1 and A.9.3 clearly hold for c so we just need to show that Definition A.9.2holds. Let ϕ , . . . , ϕ m − , ψ , . . . , ψ m ′ − ∈ G ∪ { W } and [ p ∈ S m,k \ i ≤ k ϕ p i ⊆ [ p ∈ S m ′ ,k \ i ≤ k ψ p i for every k < m . Since c a n are coherent on G ∪ { W } and thus extend to measures on an algebracontaining G ∪ { W } , we have by Corollary A.7 that m − X k =0 c a n ( ϕ k ) = m − X k =0 c a n ( [ p ∈ S m,k \ i ≤ k ϕ p i ) ≤ m ′ − X k =0 c a n ( [ p ∈ S m ′ ,k \ i ≤ k ψ p i ) = m ′ − X k =0 c a n ( ψ k ) It is a standard fact that convergence in measure implies a.e. convergence on a subsequence. Now notice thatloose convergence implies convergence in measure when the measure is finite. [ p ∈ S m,k \ i ≤ k ϕ p i = [ p ∈ S m ′ ,k \ i ≤ k ψ p i = ∅ for k ≥ m ′ . Sending n to infinity and using the pointwise convergence of c a n to c on G ∪ { W } weconclude that m − X k =0 c ( ϕ k ) ≤ m ′ − X k =0 c ( ψ k ) . Thus c is a partial measure on G ∪ { W } . By Theorem A.10, it follows that there is a finitelyadditive probability function c ∗ on A ( F ) such that c ∗ = c on G ∪ { W } . Thus c ∗ | F is a coherentcredence function on F and c = ¯ c | F = c ∗ | F µ -a.e. (specifically off G c ). Further, we already assumed c is A -measurable and { p : c ( p ) ∈ [0 , } ⊆G . Thus c is a µ -coherent µ -credence function.The proof is finished just as in the proof of Theorem 3.4. By Theorem A.5, π c is the generalizedprojection of c onto E . Since B ( E, c ) = inf s ∈ E ( s, c ) < ∞ there is a B-minimizing sequence { s n } of elements in E such that B ( s n , c ) → B ( E, c ) by thedefinition of infimum. By the definition of a generalized projection, s n µ π c . Since E is closedunder loose convergence, it follows that π c ∈ E . Further, since c is µ -incoherent we know c = π c (up to µ -a.e. equivalence) so we see B ( E, c ) ≥ B ( π c , c ) > since B ( s, t ) = 0 if and only if s = tµ -a.e. Since I ( c, w ) < ∞ for all w , we deduce that I ( c, w ) ≥ B ( E, c ) + I ( π c , w ) > I ( π c , w ) for all w ∈ W . This proves that c is strongly dominated by π c , and we are done. References
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