Information algebras in the theory of imprecise probabilities
aa r X i v : . [ c s . A I] F e b Information algebras of coherent sets ofgambles
Juerg Kohlas
Department of Informatics DIUFUniversity of FribourgCH – 1700 Fribourg (Switzerland)
Arianna Casanova, Marco Zaffalon
Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA)Lugano (Switzerland)
March 1, 2021
Abstract
In this paper, we show that coherent sets of gambles can be embeddedinto the algebraic structure of information algebra . This leads firstly, to a newperspective of the algebraic and logical structure of desirability and secondly,it connects desirability, hence imprecise probabilities, to other formalism incomputer science sharing the same underlying structure. Both the domain free and the labeled view of the information algebra of coherent sets of gambles arepresented, considering a special case of possibility space.
Contents INTRODUCTION AND OVERVIEW In a recent paper (Miranda & Zaffalon, 2020) some results about compatibility orconsistency of coherent sets of gambles or lower previsisons have been derived andit was remarked that these results were in fact results of the theory of informationor valuation algebras (Kohlas, 2003). This point of view, however was not workedout in (Miranda & Zaffalon, 2020). In this paper this issue is taken up and it isshown that coherent sets of gambles, strictly desirable sets of gambles, coherentlower and upper previsions indeed form idempotent information algebras. Like ingroup theory, certain results concerning particular groups follow from general grouptheory, so many known results about desirable gambles, lower and linear previsionsare indeed properties of an information algebra and follow from the correspondinggeneral theory. Some of these results are discussed in this paper, but there aredoubtless many other properties which can be derived from the theory of informationalgebra.From the point of view of information algebra, sets of gambles or lower previsions arepieces of information about certain groups of variables with their respective sets ofpossibilities. Such pieces of information can be aggregated or combined, and the in-formation they contain about other groups of variables can be extracted. This leadsto an algebraic structure satisfying a number of simple axioms. In fact there aretwo different versions of information algebras, a domain-free one and a labeled one.They are closely related and each one can be derived or reconstructed form the otherone. Grossly, the domain-free version is better suited for theoretical studies, since itis a structure of universal algebra, whereas the labeled version is better adapted tocomputational purposes, since it provides more efficient storage structures. In fact,labeled information (or valuation algebras, if idempotency is dropped) are the uni-versal algebraic structures for local computation in join or junction trees as originallyproposed for probabilistic networks by (Lauritzen & Spiegelhalter, 1988). Based onthis work (Shenoy & Shafer, 1990) proposed a first axiomatic scheme which wassufficient to generalize the Lauritzen-Spiegelhalter scheme to a multitude of otheruncertainty formalisms, like Dempster-Shafer belief functions, possibility theory andmany others. In (Kohlas, 2003) the algebraic theory is systematically developed andstudied. In particular, the domain-free and labeled views are presented (based orig-inally on (Shafer, 1991)). In this paper both the domain-free and the labeled viewof desirable gambles and lower previsions are presented.
INTRODUCTION AND OVERVIEW
DESIRABLE GAMBLES
Consider a set Ω of possible worlds. A gamble over this set is a bounded function f : Ω → R . We denote the set of all gambles on Ω by L (Ω), or more simply by L when there isno possible ambiguity. We also let L + (Ω) = { f ∈ L (Ω) : f ≥ , f = 0 } , or simply L + , denote the subset of non-vanishing, non-negative gambles. A coherent set ofgambles over Ω is a subset D of L (Ω) such that1. L + (Ω) ⊆ D ,2. 0 D ,3. f, g ∈ D implies f + g ∈ D ,4. f ∈ D , and λ > λ · f ∈ D .So, D is a convex cone.If D ′ is any subset of L (Ω), then E ( D ′ ) = posi ( L + (Ω) ∪ D ′ ) , is called the natural extension of the set of gambles D ′ , where posi ( D ) is defined inthe following way: posi ( D ) = r X j =1 λ j f j : f j ∈ D, λ j > , r ≥ . The natural extension of a set of gambles D , E ( D ), is coherent if and only if 0
6∈ E ( D ).Coherent sets are closed under intersection, that is they form a topless ∩ -structure.By standard order theory (Davey & Priestley, 2002), they are ordered by inclusion,intersection is meet in this order and a family of coherent sets of gambles D i , i ∈ I ,where I is an index set, have a join if they have an upper bound among coherentsets: _ i ∈ I D i = \ { D coherent : [ i ∈ I D i ⊆ D } . DESIRABLE GAMBLES E ( D ′ ), is the smallest coherent set containing D ′ , if E ( D ′ ) is coherent, E ( D ′ ) = \ { D coherent : D ′ ⊆ D } , so that _ i ∈ I D i = E ( [ i ∈ I D i )if E ( S i ∈ I D i ) is coherent. Let C (Ω) be the family of coherent sets of gambles on Ω.In view of the following section, it is convenient to add L (Ω) to C (Ω) and let Φ = C (Ω) ∪ {L (Ω) } . The family of sets in Φ is still a ∩ -structure, but now a toppedone. So, again by standard results of order theory (Davey & Priestley, 2002), Φ isa complete lattice under inclusion, meet is intersection and join is defined for anyfamily of sets D i ∈ Φ , ∀ i ∈ I , as _ i ∈ I D i = \ { D ∈ Φ : [ i ∈ I D i ⊆ D } . Note that, if a family of coherent sets D i has no upper bound in C (Ω), then its joinis simply L (Ω). In this topped ∩ -structure, C ( D ′ ) = \ { D ∈ Φ : D ′ ⊆ D } is a closure (or consequence) operator on the subsets of gambles, (Davey & Priestley, 2002),i.e., given D, D ′ ⊆ L (Ω):1. D ⊆ C ( D ),2. D ⊆ D ′ implies C ( D ) ⊆ C ( D ′ ),3. C ( C ( D )) = C ( D ).Note that C ( D ) = E ( D ), if 0
6∈ E ( D ), that is if E ( D ) is coherent. Otherwise we mayhave E ( D ) = L (Ω). We refer to (de Cooman, 2010) for a similar order-theoreticview on belief models. These results prepare the way to an information algebra ofcoherent sets of gambles (see next section). For further reference, note the followingwell-known result on consequence operators. Lemma 1 If C is a consequence operator on sets of gambles then, for any D , D ⊆L (Ω) : C ( D ∪ D ) = C ( C ( D ) ∪ D ) . Proof.
Obviously, C ( D ∪ D ) ⊆ C ( C ( D ) ∪ D ). On the other hand D , D ⊆ D ∪ D ,hence C ( D ) ⊆ C ( D ∪ D ) and D ⊆ C ( D ∪ D ). This implies C ( C ( D ) ∪ D ) ⊆C ( C ( D ∪ D )) = C ( D ∪ D ). ⊓⊔ A coherent set of gambles M is called maximal , if it is not a proper subset of acoherent set of gambles. These sets play an important role because of the followingfacts proved in (De Cooman & Quaeghebeur, 2012): DESIRABLE GAMBLES
61. Any coherent set of gambles is a subset of a maximal one,2. Any coherent set of gambles is the intersection of all maximal coherent sets itis contained in.In addition, maximal coherent sets of gambles are characterized by the followingcondition, (De Cooman & Quaeghebeur, 2012) ∀ f ∈ L − { } : f M ⇒ − f ∈ M. For a discussion of the importance of maximal coherent sets of gambles in relationto information algebras, see Section 5.A further important class of coherent sets of gambles are the strictly desirable setsof gambles . We shall employ the notation D + for strictly desirable sets of gambles,to differentiate them from the general case of coherent sets of gambles.In addition to the conditions 1. to 4. above for coherence, the following conditionis added:5 f ∈ D + implies either f ≥ f − δ ∈ D + for some δ > almost desirable setof gambles , that we indicate with the notation ¯ D , satisfying the following conditions(Walley, 1991)1. f ∈ ¯ D implies sup f ≥ f > f ∈ ¯ D ,3. f, g ∈ ¯ D implies f + g ∈ ¯ D ,4. f ∈ ¯ D and λ > λ · f ∈ ¯ D ,5. f + δ ∈ ¯ D for all δ > f ∈ ¯ D .Such a set is no more coherent since it contains f = 0.Let us indicate with ¯ C (Ω) the family of almost desirable sets of gambles. We remarkthat ¯ C (Ω) again forms a ∩ -system, still topped by L (Ω). Therefore if we add, asbefore, L (Ω) to ¯ C (Ω) then ¯Φ = ¯ C (Ω) ∪ {L (Ω) } is a complete lattice too.So, we may define the operator:¯ C ( D ′ ) = \ { ¯ D ∈ ¯Φ : D ′ ⊆ ¯ D } . that is still a closure operator on sets of gambles.So far, we have considered sets of gambles in L (Ω) relative to a fixed set of possibili-ties Ω. Next some more structure among sets of possibilities is introduced related tofamily of variables and their possible values. This is the base for relating coherentsets of gambles to information algebras. DOMAIN-FREE INFORMATION ALGEBRA We may consider a coherent set of gambles as someone’s evaluation of the “chances”of the elements of Ω as possible answers. This can be considered as a piece ofinformation about Ω. Here, in this spirit, coherent sets of gambles are shown toform an information algebra. We consider for this a special form of a domain Ω,namely a multivariate model.Let I be an index set of variables X i , i ∈ I . In practice, I is often assumed to befinite or countable. But we need not make this restriction. Any variable X i has adomain of possible values Ω i . For any subset S of I letΩ S = × i ∈ S Ω i and Ω = Ω I . We consider coherent sets of gambles on Ω, or rather Φ = C (Ω) ∪{L (Ω) } . The tuples ω in Ω can be seen as functions ω : I → Ω, so that ω i ∈ Ω i ,for any i ∈ I . A gamble f on Ω is called S -measurable, if for all ω, ω ′ ∈ Ω with ω | S = ω ′ | S we have f ( ω ) = f ( ω ′ ) (here ω | S is the restriction of the map ω to S ). Let L S denote the set of all S -measurable gambles. Define L ∅ = R , the set of constantgambles, and note that L I = L (Ω). For further reference we have the followingresult. Lemma 2
For any subsets S and T of I , L S ∩ T = L S ∩ L T . Proof.
If one of the two subsetes is empty we immediatly have the result. Otherwise,assume first f ∈ L S ∩ T . Consider two elements ω, µ ∈ Ω so that ω | S = µ | S . Then wehave also ω | S ∩ T = µ | S ∩ T and f ( ω ) = f ( µ ). So we see that f ∈ L S and similarly f ∈ L T .Conversely, assume f ∈ L S ∩ L T and consider two tuples so that ω | S ∩ T = µ | S ∩ T .Define then a tuple λ by λ i = ω i = µ i , i ∈ S ∩ T,ω i , i ∈ S − T,µ i , i ∈ T − S, , i ∈ ( S ∪ T ) c . Then λ | S = ω | S and λ | T = µ | T . Since f is both S - and T -measurable we have f ( ω ) = f ( µ ). It follows that f ∈ L S ∩ T , and this concludes the proof. ⊓⊔ Now we define in Φ two operations. Let
D, D , D ∈ Φ and S ⊆ I :1. Combination: D · D = D ∨ D = C ( D ∪ D ),2. Extraction: ǫ S ( D ) = C ( D ∩ L S ). DOMAIN-FREE INFORMATION ALGEBRA E S ( D ) = C ( D ) ∩ L S , for any set of gambles D . Then, if D is coherent, we have: ǫ S ( D ) = C ( E S ( D )) . If we consider coherent sets of gambles as pieces of information (about someonesbelief on possible answers in Ω) then combination represents the aggregation of twopieces of information and extraction the coarsening or extraction of an informationto a subset of variables. Note that D · D = L (Ω), for some D , D ∈ Φ, meansthat the two sets D and D are not compatible, that is, D ∪ D is not coherent.So, clearly L (Ω) is the null element of combination and represents contradiction orincompatibility. The set L + (Ω) is the unit element of combination, representingvacuous information. We claim that Φ equipped with these two operations satisfiesthe properties collected in the following theorem. Theorem 1 (Φ; · ) is a commutative semigroup with a null and unit ,2. for any subset S ⊆ I and D, D , D ∈ Φ : E1 ǫ S (0) = 0 , E2 ǫ S ( D ) · D = D , E3 ǫ S ( ǫ S ( D ) · D ) = ǫ S ( D ) · ǫ S ( D ) .3. for any S, T ⊆ I , ǫ S ◦ ǫ T = ǫ T ◦ ǫ S = ǫ S ∩ T .Proof. That (Φ , · ) is a commutative semigroup follows from D · D = D ∨ D , forany D , D in the complete lattice Φ. As stated above, 0 = L (Ω) is the null elementand 1 = L + (Ω) the unit element of the semigroup (null and unit in a semigroup arealways unique). This proves item 1.For E1 we have ǫ S (0) = ǫ S ( L (Ω)) = C ( L (Ω) ∩ L S ) = C ( L S ) = L (Ω) = 0 . for any S ⊆ I .E2 follows since D ∩ L S ⊆ D and C ( D ∩ L S ) ⊆ D , for any D ∈ Φ, S ⊆ I . To proveE3 define, using Lemma 1 A = C ( C ( D ∩ L S ) ∪ D ) ∩ L S = C (( D ∩ L S ) ∪ D ) ∩ L S B = C ( C ( D ∩ L S ) ∪ ( C ( D ∩ L S )) = C (( D ∩ L S ) ∪ ( D ∩ L S )) . Then we have B = ǫ S ( D ) · ǫ S ( D ) and C ( A ) = ǫ S ( ǫ S ( D ) · D ). Note that B ⊆ C ( A ).We claim first that: ǫ S ( D ) · ǫ S ( D ) = 0 ⇐⇒ ǫ S ( D ) · D = 0 . (3.1)Indeed, ǫ S ( D ) · ǫ S ( D ) = 0 implies a fortiori ǫ S ( D ) · D = 0. DOMAIN-FREE INFORMATION ALGEBRA ǫ S ( D ) · D = 0. This equivalently means that 0 ∈ C ( C ( D ∩L S ) ∪ D ) = C (( D ∩ L S ) ∪ D ), by Lemma 1. Now, two cases are possible. If D = L (Ω) or D = L (Ω) we have immediately the result, otherwise we claim that0 = f + g ′ with f ∈ D ∩ L S and g ′ ∈ D ∩ L S . Indeed, from 0 ∈ C (( D ∩ L S ) ∪ D ),we know that 0 = f + g + h ′ with f ∈ D ∩ L S , g ∈ D , h ′ ∈ L + (Ω) ⊆ D . Then, ifwe introduce g ′ = g + h ′ , we have 0 = f + g ′ with f ∈ D ∩ L S , g ′ ∈ D . However,this implies g ′ = − f ∈ L S and then g ′ ∈ D ∩ L S .Notice that ǫ S ( D ) · ǫ S ( D ) = B = C (( D ∩ L S ) ∪ ( D ∩ L S )). Therefore, we havethe result.So, we may now assume that both ǫ S ( D ) · D and ǫ S ( D ) · ǫ S ( D ) are coherent.Then we have A = { f ∈ L S : f ≥ λg + µh, g ∈ D ∩ L S , h ∈ D , λ, µ ≥ , f = 0 } . Consider f ∈ A . Then f = λg + µh + h ′ , where h ′ ∈ L + (Ω). Since f and g are S -measurable, µh + h ′ must be S -measurable, hence µh + h ′ ∈ D ∩ L S . But thissays that f ∈ B , hence we have C ( A ) ⊆ C ( B ) = B .To prove item 3, note first that ǫ S ◦ ǫ T ( D ) = 0 and ǫ S ∩ T ( D ) = 0 if and only if D = 0.So assume D to be coherent. Then we have ǫ S ( ǫ T ( D )) = C ( C ( D ∩ L T ) ∩ L S ) ,ǫ S ∩ T ( D ) = C ( D ∩ L S ∩ T ) = C ( D ∩ L T ∩ L S ) . Obviously, ǫ S ∩ T ( D ) ⊆ ǫ s ( ǫ T ( D )). Consider then f ∈ C ( D ∩ L T ) ∩ L S , so that f ∈ L S , f ≥ g, g ∈ D ∩ L T . Define g ′ ( ω ) = sup λ | S = ω | S g ( λ ) . Then we have f ≥ g ′ . Clearly g ′ is S -measurable and belongs to D , g ′ ∈ D ∩ L S .We claim that g ′ is also T -measurable. Consider two elements ω and µ so that ω | S ∩ T = µ | S ∩ T . Note that we may write g ′ ( ω ) = sup λ | S = ω | S g ( λ ) = sup λ | I − S g ( ω | S ∩ T, ω | S − T, λ | T − S, λ | R ) , where R = ( S ∪ T ) c . Similarly, we have g ′ ( µ ) = sup λ ′ | S = µ | S g ( λ ′ ) = sup λ ′ | I − S g ( ω | S ∩ T, µ | S − T, λ ′ | T − S, λ ′ | R ) , Since g is T -measurable, we have: g ′ ( µ ) = sup λ ′ | S = µ | S g ( λ ′ ) = sup λ ′ | I − S g ( ω | S ∩ T, ω | S − T, λ ′ | T − S, λ ′ | R ) , DOMAIN-FREE INFORMATION ALGEBRA g ′ ( ω ).This shows that g ′ is S ∩ T -measurable, therefore both S - and T -measurable, byLemma 2. So we have g ′ ∈ D ∩ L S ∩ L T , hence f ∈ C ( D ∩ L T ∩ L S ). This provesitem 3. ⊓⊔ A system with two operations satisfying the conditions of Theorem 1 is called adomain-free information algebra (Kohlas, 2003). There is an alternative, equivalentversion, a so-called labeled information algebra, which may derived from it, see thenext section. In any information algebra, an information order can be introduced. Inthe case of coherent sets of gambles, this order is defined as D ≤ D if D · D = D . D is more informative than D if adding D gives nothing new; D is alreadycontained in D . It is easy to verify that it is a partial order and also that D ≤ D ifand only if D ⊆ D . Moreover, combination D · D yields the supremum, D · D =sup { D , D } . This is also written as the join D · D = D ∨ D . The vacuousinformation L + (Ω) is the least information in this order and the inconsistency L (Ω)is the top element (although strictly speaking it is no more an information). Thismeans that (Φ , ≤ ) is a join semi-lattice under information order; in fact it is acomplete lattice, since information order corresponds to set inclusion. ConditionsE1 to E3 in Theorem 1 can also be rewritten using this order as the following: forany subset S ⊆ I and D, D , D ∈ Φ, E1 ǫ S (0) = 0, E2 ǫ S ( D ) ≤ D , E3 ǫ S ( ǫ S ( D ) ∨ D ) = ǫ S ( D ) ∨ ǫ S ( D ).In algebraic logic such an operator is also called an existential quantifier .We further claim that extraction distributes over intersection (or meet in the com-plete lattice). Theorem 2
Let D j , j ∈ J be any family of sets of gambles in Φ and S any subsetof variables. Then ǫ S ( \ j ∈ J D j ) = \ j ∈ J ǫ S ( D j ) . (3.2) Proof.
We may assume that D j ∈ C (Ω) ∀ j ∈ J , since if some or all D j = L (Ω),then we may restrict the intersection on both sides over the set D j ∈ C (Ω), or theintersection over both sides equals L (Ω). If D j ∈ C (Ω) , ∀ j ∈ J , we have ǫ S ( \ j ∈ J D j ) = E (( \ j D j ) ∩ L S ) = posi ( L + (Ω) ∪ (( \ j D j ) ∩ L S )) \ j ∈ J ǫ S ( D j ) = \ j E ( D j ∩ L S ) = \ j posi ( L + (Ω) ∪ ( D j ∩ L S )) . Although usually operators on a Boolean lattice are considered and the order is inverse to theinformation order.
LABELED INFORMATION ALGEBRA f ∈ ǫ S ( T j ∈ J D j ), so that f = λg + µh , where λ, µ nonnegativeand not both equal zero and g ∈ ( T j D j ) ∩ L S and h ∈ L + (Ω). But then g ∈ D j ∩ L S for all j, so that f ∈ T j ∈ J ǫ S ( D j ).Conversely, assume f ∈ T j ∈ J ǫ S ( D j ). Then f ≥ g j for some g j ∈ ( D j ∩ L S ) , ∀ j .Hence, f ( ω ) ≥ sup k g k ( ω ), for every ω ∈ Ω. However, sup k g k ∈ ∩ j ( D j ) becausesup k g k ( ω ) ≥ g j ( ω ) , ∀ j, ∀ ω . Moreover, sup k g k ∈ L S because, thanks to the fact thatevery g j ∈ L S , we have the following: if ω | S = ω ′ | S , then sup k g k ( ω ) = sup k g k ( ω ′ ).Therefore f ∈ ǫ S ( ∩ j D j ). ⊓⊔ So (Φ , ≤ ) is a lattice under information order and satisfies (3.2). Such an informationalgebra is called a lattice information algebra.The family of strictly desirable sets of gambles D + is also closed under combination(if incompatibility is admitted) and extraction in Φ. Therefore, if the null element L (Ω) is added and we denote with Φ + the union of all the strictly desirable sets ofgambles and L (Ω), we have that Φ + forms a subalgebra of Φ. There is a general method to derive a corresponding labeled information algebrafrom a domain-free information algebra (Kohlas, 2003). We describe it here in theframe of the information algebra of coherent sets of gambles. In the particular case ofcoherent sets of desirable gambles, as well as in the case of lower previsions, there isa second, isomorphic version of the labeled algebra, which is nearer to the intuitionand which will be introduced after the general construction of labeled algebras.From a labeled algebra the domain-free information algebra may be reconstructed(Kohlas, 2003). So the the two views of information algebras are equivalent. Thewhole theory presented here could also have been developed in the labeled view. Itis a question of convenience whether the domain-free or the labeled view is chosen.Usually, for theoretical considerations, the domain-free view is preferable, because itis nearer to universal algebra. For computational purposes, the labeled view is usedin general, because it corresponds better to the needs of efficient data representation.A subset S of I is called support of a coherent set of gambles D , if ǫ S ( D ) = D .This means that the information concerns or is focused to the group S of variables.Here are a few well-known results on support in domain-free information algebras,let D, D , D ∈ Φ , S, T ⊆ I :1. Any S is a support of the null element 0 ( L (Ω)) and the unit 1 ( L + (Ω)) S ofvariables,2. S is a support of ǫ S ( D ),3. if S is a support of D , then S is a support of ǫ T ( D ),4. if S and T are supports of D , then so is S ∩ T , LABELED INFORMATION ALGEBRA S is a support of D , then ǫ T ( D ) = ǫ S ∩ T ( D ),6. If S is a support of D and S ⊆ T , then T is a support of D ,7. if S is a support of D and D , then it is also a support of D · D ,8. if S is a support of D and T a support of D , then S ∪ T is a support of D · D .For proofs see (Kohlas, 2003; Kohlas, 2017). Now, we consider sets Ψ S of pairs( D, S ), where S ⊆ I is a support of D ∈ Φ. That is, we collect together pieces ofinformation concerning the same set of variables. LetΨ = [ S ⊆ I Ψ S . In Ψ we define the following operations in terms of the operations of the domain-freealgebra, let (
D, S ) , ( D , S ) , ( S , T ) ∈ Ψ:1. Labeling: d ( D, S ) = S ,2. Combination: ( D , S ) · ( D , T ) = ( D · D , S ∪ T ),3. Projection (Marginalization): π T ( D, S ) = ( ǫ T ( D ) , T ) defined for T ⊆ S ⊆ I .It is well-known and easy to verify, that Ψ with these three operations satisfies theproperties in the following theorem (Kohlas, 2003). Theorem 3
Semigroup: (Ψ , · ) is a commutative semigroup;Labeling: d (( D , S ) · ( D , T )) = d ( D , S ) ∪ d ( D , T ) , for any ( D , S ) , ( D , T ) ∈ Ψ and d ( π T ( D, S )) = T , for any ( D, S ) ∈ Ψ , T ⊆ S ⊆ I ;Null and Unit: (0 , S ) · ( D, S ) = (0 , S ) , (1 , S ) · ( D, S ) = (
D, S ) for any ( D, S ) ∈ Ψ , π T ( D, S ) = (0 , T ) if and only if ( D, S ) = (0 , S ) , π T (1 , S ) = (1 , T ) , with T ⊆ S ⊆ I and (1 , S ) · (1 , T ) = (1 , S ∪ T ) , with S, T ⊆ I ;Transitivity of Projection: if R ⊆ S ⊆ T ⊆ I , then π R ( D, T ) = π R ( π S ( D, T )) , forany ( D, T ) ∈ Ψ ;Combination: π S (( D , S ) · ( D , T )) = ( D , S ) · π S ∩ T ( D , T ) , for any ( D , S ) , ( D , T ) ∈ Ψ ;Idempotency: ( D, S ) · π T ( D, S ) = (
D, S ) , for any ( D, S ) ∈ Ψ , T ⊆ S ⊆ I ;Identity: π S ( D, S ) = (
D, S ) , for any ( D, S ) ∈ Ψ . LABELED INFORMATION ALGEBRA S of variables, let C S (Ω) be the family of sets of gambles coherent relative to L S , thatis sets D ∩ L S , where D ∈ C (Ω). Further let ˜Ψ S be C S (Ω) ∪ {L S } and˜Ψ = [ S ⊆ I ˜Ψ S . Within ˜Ψ we define the following operations, let ( D ∩ L S , S ) , ( D ∩ L S , S ) , ( D ∩L T , T ) ∈ ˜Ψ:1. d ( D ∩ L S , S ) = S ,2. Combination: ( D ∩ L S , S ) · ( D ∩ L T , T ) = (( D · D ) ∩ L S ∪ T , S ∪ T ), where D · D = C ( D ∪ D ) is the domain-free combination,3. Projection (Marginalization): π T ( D ∩ L S , S ) = ( D ∩ L T , T ) defined if T ⊆ S .Consider now the map h : Ψ → ˜Ψ defined by ( D, S ) ( D ∩ L S , S ) and recall thathere D = ǫ S ( D ). The map h will establish an isomorphism between the labeledinformation algebra Ψ and ˜Ψ. Theorem 4
The map h has the following properties:1. It maintains combination, null and unit, and projection. Let ( D, S ) , ( D , S ) , ( D , T ) ∈ Ψ : h (( D , S ) · ( D , T )) = h ( D , S ) · h ( D , T ) ,h ( L (Ω) , S ) = ( L S , S ) ,h ( L + (Ω) , S ) = ( L + S , S ) ,h ( π T ( D, S )) = π T ( h ( D, S )) , if T ⊆ S. h is bijective.Proof.
1. we have, by definition h (( D , S ) · ( D , T )) = h ( D · D , S ∪ T ) = (( D · D ) ∩ L S ∪ T , S ∪ T )= ( D ∩ L S , S ) · ( D ∩ L T , L T ) = h ( D , S ) · h ( D , T ) . Obviously, ( L (Ω) , S ) maps to ( L S , S ) and ( L + (Ω) , S ) maps to ( L + S , S ). And thenwe have, again by definition h ( π T ( D, S )) = h ( ǫ T ( D ) , T ) = ( ǫ T ( D ) ∩ L T , T ) . ATOMS AND MAXIMAL COHERENT SETS OF GAMBLES ǫ T ( D ) ∩ L T = D ∩ L T . Clearly ǫ T ( D ) ∩ L T ⊆ D ∩ L T and, onthe other hand, D ∩ L T ⊆ C ( D ∩ L T ) ∩ L T = ǫ T ( D ) ∩ L T . So we obtain h ( π T ( D, S )) = ( ǫ T ( D ) ∩ L T , T ) = ( D ∩ L T , T ) = π T ( D ∩ L S , S ) = π T ( h ( D, S )) . This proves item 1.2. Suppose h ( D , S ) = h ( D , T ). Then first we must have S = T and further,since then D ∩ L S = D ∩ L S , we have D = ǫ S ( D ) = C ( D ∩ L S ) = C ( D ∩L S ) = ǫ S ( D ) = D . So the map h is injective. And for any ( D ∩ L S , S ) we have h ( ǫ S ( D ) , S ) = ( ǫ S ( D ) ∩ L S , S ) = ( D ∩ L S , S ) by the first part of the proof. So h issurjective, hence bijective. ⊓⊔ We remark, that in labeled information algebras, as in domain-free ones, an infor-mation order can be defined by ( D ∩ L S , S ) ≤ ( D ∩ L T , T ) if( D ∩ L S , S ) · ( D ∩ L T , T ) = ( D ∩ L T , T ) . By definition of combination this means that S ⊆ T and D · D = D , hence D ⊆ D .In a computational application of this second version of the labeled informationalgebra, one would use the fact that any set D ∩ L S is determined by the valuesof the gambles f on the set of possibilities Ω S , which reduce greatly the efficiencyof storage. Observations like this one explain why labeled information algebra arebetter suited for computational purposes. In certain information algebras there are maximally informative elements, called atoms (Kohlas, 2003). This is in particular the case for the information algebra ofcoherent sets of gambles. Maximal coherent sets of gambles M (see Section 2) havethe property that, in information order, M ≤ D for D ∈ Φ ⇒ M = D or D = L (Ω)Elements in an information algebra with this property are called atoms (Kohlas, 2003).In certain cases atoms determine the structure of an information algebra fully. Weshall show that this is indeed the case for the algebra of coherent sets of gambles,see in particular the next Section 6. The definition of an atom can alternativelyaccording to the definition of information order also be expressed by combination, M is an atom if M · D = L (Ω) or M · D = M, ∀ D ∈ Φ . Let At (Ω) denote the set of all atoms (maximal coherent sets) on Ω. For any coherentset of gambles D , let At ( D ) denote the set of atoms (maximal coherent sets) whichcontain D , At ( D ) = { M ∈ At (Ω) : D ≤ M } . ATOMS AND MAXIMAL COHERENT SETS OF GAMBLES D of gambles, D ∈ C (Ω), there is a maximal set (anatom) M so that in information order D ≤ M (i.e D ⊆ M ). So At ( D ) is neverempty. An information algebra with this property is called atomic .2. For all coherent sets of gambles D ∈ C (Ω), we have D = inf At ( D ) = \ At ( D ) . An information algebra with this property is called atomic composed (Kohlas, 2003)or atomistic .3. For any, not empty, subset A of At (Ω) we have thatinf A = \ A is a coherent set of gambles, i.e. an element of C (Ω). Such an algebra is called completely atomistic .The first two properties are proved in (De Cooman & Quaeghebeur, 2012), the thirdfollows since coherent sets form a T -structure. Note that, if A is a set of maximalsets of gambles, A ⊆ At ( T A ), and in general A is a proper subset of At ( T A ).These properties determine the structure of the information algebra of coherent setsin term of so-called set algebras, as we shall discuss in the following Section 6.A consideration of the labeled version ˜Ψ of the information algebra of coherentsets of gambles, exhibits some further structure of atoms. In fact, we may havemaximally informative elements relative to a domain Ω S for any subset of variables.If M ∈ At (Ω), then we have by definition of atom:( D ∩ L S , S ) ≥ ( M ∩ L S , S ) for some D ∈ Φ , if and only if D = M or D = L (Ω)In other words, we have ( D ∩ L S , S ) ≥ ( M ∩ L S , S ) if and only if either ( D ∩ L S , S ) =( M ∩ L S , S ) or ( D ∩ L S , S ) = ( L S , S ). This means that the elements ( M ∩ L S , S are maximally informative relative to the subset of variables S . Such elements arecalled atoms relative to S (Kohlas, 2003). Such relative atoms have the followingproperties. Lemma 3
Assume M and M , M to be atoms, D ∈ Φ , S ⊆ I . Then1. ( M ∩ L S , S ) · ( D ∩ L S , S ) = ( L S , S ) or = ( M ∩ L S , S ) .2. If T ⊆ S , then π T ( M ∩ L S , S ) is an atom relative to T .3. Either ( D ∩ L S , S ) ≤ ( M ∩ L S , S ) or ( D ∩ L S , S ) · ( M ∩ L S , S ) = ( L S , S ) . SET ALGEBRAS
4. Either ( M ∩ L S , S ) · ( M ∩ L S , S ) = ( L S , S ) or ( M ∩ L S , S ) = ( M ∩ L S , S ) . These are purely properties of information algebras, for a proof see (Kohlas, 2003).The properties of the algebra of being atomic, atomistic and completely atomisticcarry over the the labeled version of the algebra of coherent sets.1.
Atomic:
For any element ( D ∩ L S , S ) ∈ ˜Ψ S , S ⊆ I there is a relative atom( M ∩ L S , S ) so that ( D ∩ L S , S ) ≤ ( M ∩ L S , S ).2. Atomistic: For any element ( D ∩ L S , S ) ∈ ˜Ψ S , S ⊆ I , ( D ∩ L S , S )) = inf { ( M ∩L S , S ) : M ∈ At ( D ) } .3. Completly Atomistic: for any, not empty, subset A of At (Ω), inf { ( M ∩ L S , S ) : M ∈ A } exists and belongs to ˜Ψ S , for every S ⊆ I .As in the domain-free case these properties imply that the atoms, the maximal coher-ent sets of gambles, determine the structure of the information algebra of coherentgambles. This will be discussed in the next section. Important instances of information algebras are set algebras . Its elements are subsetsof Ω. The operation of combination is simply set intersection. Extraction is definedin term of cylindrification: if A is a subset of Ω, then its cylindrification with respectto a subset S of variables is defined as σ S ( A ) = { ω ∈ Ω : ∃ ω ′ ∈ A so that ω ′ | S = ω | S } . This is a saturation operator. The family of subsets of Ω with intersection ascombination and cylindrification as extraction is a domain-free information algebra(Kohlas, 2017). Saturation operators are more generally defined relative to parti-tions or equivalence relations of a set. In the present case we have the relations ω ≡ S ω ′ with ω, ω ′ ∈ Ω , S ⊆ I , if ω | S = ω ′ | S . Below we shall encounter a moregeneral case.We show now that such a set algebra can be embedded into the information algebraof coherent sets of gambles. Define for any subset A not empty of Ω D A = { f ∈ L (Ω) : inf ω ∈ A f ( ω ) > } ∪ L + (Ω) . This is obviously a coherent set of gambles. And we define D ∅ = L (Ω).Now we have the following result. Theorem 5
For all subsets A and B of Ω and subsets S of I D ∅ = L (Ω) , D Ω = L + (Ω) , SET ALGEBRAS D A · D B = D A ∩ B ,3. ǫ S ( D A ) = D σ S ( A ) .Proof. D A = L + or D B = L + then A = Ω or B = Ω, hence we have the result.Now, suppose D A , D B = L + . If A ∩ B = ∅ , then D A ∩ B = L (Ω).If A = ∅ or B = ∅ then we have the result. Consider instead the case in which A, B = ∅ . By definition we have D A · D B = C ( D A ∪ D B ). Consider f ∈ D A \ L + and g ∈ D B \ L + . Since A and B are disjoint, we have ˜ f ∈ D A and ˜ g ∈ D B , where˜ f , ˜ g are defined in the following way:˜ f ( ω ) = f ( ω ) for ω ∈ A, − g ( ω ) for ω ∈ B, ω ∈ ( A ∪ B ) c ˜ g ( ω ) = − f ( ω ) for ω ∈ A,g ( ω ) for ω ∈ B, ω ∈ ( A ∪ B ) c But then ˜ f + ˜ g = 0 ∈ D A · D B , hence D A · D B = L (Ω) = D A ∩ B .Assume then that both A ∩ B = ∅ . Note that D A ∪ D B ⊆ D A ∩ B so that D A · D B iscoherent and D A · D B ⊆ D A ∩ B . Consider then a gamble f ∈ D A ∩ B . If inf A f > f ∈ D A and if inf B f >
0, then f ∈ D B and in both case we have f ∈ D A ∪ D B ,hence in D A · D B . Otherwise select a δ > f ( ω ) = / f ( ω ) for ω ∈ A ∩ B, ω ∈ ( A ∪ B ) c ,δ for ω ∈ A − B.f ( ω ) − δ for ω ∈ B − Af ( ω ) = / f ( ω ) for ω ∈ A ∩ B, ω ∈ ( A ∪ B ) c ,f ( ω ) − δ for ω ∈ A − B.δ for ω ∈ B − A Then, f = f + f and f ∈ D A , f ∈ D B . Therefore f ∈ C ( D A ∪ D B ) = D A · D B ,hence D A · D B = D A ∩ B .3.) If A is empty, then ǫ S ( D ∅ ) = L (Ω) so that item 3 holds in this case. So assume A = ∅ . Then we have ǫ S ( D A ) = C ( D A ∩ L S ) = posi ( L + (Ω) ∪ ( D A ∩ L S )) . Consider then a gamble f ∈ D A ∩ L S . Then, we have inf A f > f is S -measurable. So, if ω | S = ω ′ | S for some ω ′ ∈ A and ω ∈ Ω, then f ( ω ) = f ( ω ′ ).Therefore inf σ S ( A ) f = inf A f >
0, hence f ∈ D σ S ( A ) . Conversely, consider a gamble f ∈ D σ S ( A ) . D σ S ( A ) is a strictly desirable set of gambles, hence, if f ∈ D σ S ( A ) , SET ALGEBRAS f ∈ L + (Ω) or there exists δ > f − δ ∈ D σ S ( A ) . If f ∈ L + (Ω), then f ∈ ǫ S ( D A ). Otherwise, let us define for every ω ∈ Ω, g ( ω ) = inf ω ′ | S = ω | S f ( ω ′ ) − δ. If ω ∈ A , then g ( ω ) > σ S ( A ) ( f − δ ) >
0. So we have inf A g ≥ g is S -measurable. However, inf A ( g + δ ) = inf A g + δ > g + δ ) ∈ D A ∩ L S and f ≥ g + δ . Therefore f ∈ C ( D A ∩ L S ). This concludes the proof. ⊓⊔ This theorem shows that the map A D A is a homomorphism between the setalgebra and the information algebra of coherent sets of gambles. Furthermore, themap is one-to-one, hence an embedding of the set algebra into the algebra of coherentsets of gambles. This is a manifestation of the fact that the theory of desirablegambles covers among other things propositional and predicate logic.Recall that coherent sets of gambles form a lattice (Section 2) where meet is setintersection. This is also the case for subsets of Ω; they form even a Boolean lattice.We need however to stress that in information order, A ≤ B iff B ⊆ A . That is,information order is the opposite of the usual inclusion order between sets. Thismeans that in information order meet is set union. Given this observation, it turnsout that the map A D A is even a lattice homomorphism. Theorem 6
For all subsets A and B of Ω , D A ∩ D B = D A ∪ B . Proof.
A gamble f belongs to D A ∩ D B if and only if both inf A f and inf B f areboth positive. But then it belongs to D A ∪ B . ⊓⊔ But there is much more about set algebras and information algebras of coherentsets of gambles. And this depends on the atomisticity of the information algebraof coherent sets of gambles. We stick in our discussion here to the domain-freeview. The labeled view of what follows has been described in (Kohlas, 2003). Inthe domain-free case the result we prove below states that Φ is isomorphic to a setalgebra.Consider the set of all atoms, that is all maximal coherent sets At (Φ) and defineequivalence relations M ≡ S M ′ if ǫ S ( M ) = ǫ S ( M ′ ) in At (Φ) for all subsets ofvariables S ⊆ I . Associated with these equivalence relations are saturation operators σ S defined by σ S ( X ) = { M ∈ At (Φ) : ∃ M ′ ∈ X so that M ≡ S M ′ } for any subset X of At (Φ) and S ⊆ I . Any saturation operator satisfies a numberof important properties which are related to information algebras. Lemma 4
Let σ S be a saturation operator on At (Φ) for some S ⊆ I , associatedwith the equivalence relation ≡ S , and X, Y subsets of At (Φ) . Then SET ALGEBRAS σ S ( ∅ ) = ∅ ,2. X ⊆ σ S ( X ) ,3. σ S ( σ S ( X ) ∩ Y ) = σ S ( X ) ∩ σ S ( Y ) ,4. X ⊆ Y implies σ S ( X ) ⊆ σ S ( Y ) ,5. σ S ( σ S ( X )) = σ S ( X ) ,6. X = σ S ( X ) and Y = σ S ( Y ) imply X ∩ Y = σ S ( X ∩ Y ) .Proof. Item 1., 2., 4. and 5. are obvious.For 6. Consider M ∈ σ S ( X ∩ Y ). Then there is a M ′ ∈ X ∩ Y so that M ≡ S M ′ .In particular, M ′ ∈ X , hence M ∈ σ S ( X ). At the same time, M ′ ∈ Y , hence M ∈ σ S ( Y ). Then M ∈ σ S ( X ) ∩ σ S ( Y ) = X ∩ Y . By 2. we must then have equality.For 3. observe that σ S ( X ) ∩ Y ⊆ σ S ( X ) ∩ σ S ( Y ), so that σ S ( σ S ( X ) ∩ Y ) ⊆ σ S ( σ S ( X ) ∩ σ S ( Y )) = σ S ( X ) ∩ σ S ( Y ) by 4. and 6. For the reverse inclusion notethat M ∈ σ S ( X ) ∩ σ S ( Y ) means that there are M ′ ∈ X , M ′′ ∈ Y so that M ≡ S M ′ and M ≡ S M ′′ . By transitivity we have then M ′ ≡ S M ′′ so that M ′′ ∈ σ S ( X ).Then M ≡ S M ′′ and M ′′ ∈ σ S ( X ) ∩ Y imply M ∈ σ S ( σ S ( X ) ∩ Y ). This concludesthe proof ⊓⊔ The first three items of the theorem correspond to the properties E1 to E3 of anexistential quantifier in an information algebra, if combination is intersection. Thisis a first step to show that the subsets of At (Φ) indeed form an information algebrawith intersection as combination and saturation operators σ S for S ⊆ I as extractionoperator. The missing item will be verified below.The combination of two equivalence relations is defined as ≡ S · ≡ T = { ( M, M ′ ) ∈ At (Φ) × At (Φ) : ∃ M ′′ ∈ At (Φ) , so that M ≡ S M ′′ ≡ T M ′ } In general this is no more an equivalence relation. In our case however it is anequivalence and the relations commute as the following lemma shows
Lemma 5
For the equivalence relations ≡ S and ≡ T in At (Φ) , S, T ⊆ I we have ≡ S · ≡ T = ≡ T · ≡ S = ≡ S ∩ T . Proof.
Let (
M, M ′ ) ∈≡ S · ≡ T so that there is an M ′′ such that ǫ S ( M ) = ǫ S ( M ′′ )and ǫ T ( M ′ ) = ǫ T ( M ′′ ). It follows that ǫ S ∩ T ( M ) = ǫ T ( ǫ S ( M )) = ǫ T ( ǫ S ( M ′′ ) = ǫ S ∩ T ( M ′′ ). Similarly we obtain ǫ S ∩ T ( M ′ ) = ǫ S ∩ T ( M ′′ ). But this shows that ( M, M ′ ) ∈≡ S ∩ T .Conversely, suppose ( M, M ′ ) ∈≡ S ∩ T , that is ǫ S ∩ T ( M ) = ǫ S ∩ T ( M ′ ). We claim that ǫ S ( M ) · ǫ T ( M ′ ) = 0. If, on the contrary ǫ S ( M ) · ǫ T ( M ′ ) = 0, then ǫ S ( ǫ S ( M ) · ǫ T ( M ′ )) = ǫ S ( M ) · ǫ S ∩ T ( M ′ ) = 0 and further ǫ S ∩ T ( ǫ S ( M ) · ǫ S ∩ T ( M ′ )) = ǫ S ∩ T ( M ) · SET ALGEBRAS ǫ S ∩ T ( M ′ ) = ǫ S ∩ T ( M ) = 0. But since M is an atom, this is a contradiction. So thereis an atom M ′′ ∈ At ( ǫ S ( M ) · ǫ T ( M ′ )) so that ǫ S ( M ) · ǫ T ( M ′ ) ≤ M ′′ . Then ǫ S ( M ′′ ) ≥ ǫ S ( M ) · ǫ S ∩ T ( M ′ ) = ǫ S ( M ). Therefore ǫ S ( M ′′ ) = ǫ S ( M ′′ ) · ǫ S ( M ) = ǫ S ( ǫ S ( M ′′ ) · M ) = ǫ S ( M ) since M is an atom. In the same way, we deduce ǫ T ( M ′′ ) = ǫ T ( M ′ ). But thismeans that ( M, M ′ ) ∈ ≡ S · ≡ T and we have proved ≡ S · ≡ T = ≡ S ∩ T . The otherequality follows by symmetry. ⊓⊔ As a corollary, it follows that the associated saturation operators commute.
Lemma 6
For the equivalence relations ≡ S and ≡ T in At (Φ) , S, T ⊆ I we have σ S ◦ σ T = σ T ◦ σ S = σ S ∩ T . Proof.
For any subset X of At (Φ) we have σ S ◦ σ T ( X ) = { M ∈ At (Φ) : ∃ M ′ ∈ X, ∃ M ′′ ∈ At (Φ) so that M ≡ S M ′′ ≡ T M ′ } = { M ∈ At (Φ) : ∃ M ′ ∈ X so tat M ≡ S · ≡ T M ′ } = { M ∈ At (Φ) : ∃ M ′ ∈ X so that M ≡ S ∩ T M ′ } = σ S ∩ T ( X ) . This proves that σ S ◦ σ T = σ S ∩ T . The remaining equality follows by symmetry. ⊓⊔ By this result, we have established that the power set of At (Φ) with intersectionas combination and saturation as extraction satisfies all items of Theorem 1. Thismeans that the power set P ( At (Φ)) is a domain-free information algebra. Since itselements are subsets and combination and extraction are set operations, it is a setalgebra . But in addition P ( At (Φ)) is also a complete, atomic Boolean lattice underinclusion, which corresponds to information order in P ( At (Φ)).Now in the following theorem we show that the pair of maps D At ( D ) and ǫ S σ S represents an information algebra homomorphism and also maintains ar-bitrary joins. Recall that in information order in Φ we have D ≤ D if and only if D ⊆ D . So information order is inclusion. In P ( At (Φ)) combination is intersec-tion so the information order is X ≤ Y if X ∩ Y = Y , hence Y ⊆ X . Therefore,information order in P ( At (Φ)) is the inverse of inclusion, join is in this order in-tersection, meet union. Remark that the following theorems are purely results ofatomistic information algebras and not specific to the algebra of coherent sets ofgambles. Part of the following has been developed in (Kohlas & Schmid, 2020). Theorem 7
The map D At ( D ) and ǫ S σ S , are respectively injective andbijective. Moreover, if D , D , D belong to Φ and S is a subset of I , the followingare valid1. At ( D · D ) = At ( D ) ∩ At ( D ) ,2. At ( L + (Ω)) = At (Φ) and At ( L (Ω)) = ∅ ,3. At ( ǫ S ( D )) = σ S ( At ( D )) . SET ALGEBRAS Proof.
The map ǫ S σ S is bijective by construction. The map D At ( D ) isone-to-one since Φ is atomistic.For 2, by definition we have At ( L (Ω)) = ∅ and since L + (Ω) ≤ M for all atoms M , At ( L + (Ω)) = At (Φ) (recall 0 = L (Ω) and 1 = L + (Ω)). So, if D · D = 0,then At ( D ) ∩ At ( D ) = ∅ , since otherwise there would by an atom M such that D · D ≤ M . Assume therefore D · D = 0. The since Φ is atomic, there isan atom M ∈ At ( D · D ) and since D , D ≤ D · D ≤ M , we conclude that M ∈ At ( D ) ∩ At ( D ). On the other hand, if M ∈ At ( D ) ∩ At ( D ), then D ≤ M and D ≤ M , hence D · D = D ∨ D ≤ M and therefore M ∈ At ( D · D ) Thisproves item 1.For 3, if D = 0 we have immediately the result. Otherwise, consider first an atom M ∈ σ S ( At ( D )), assuming D = 0. Then there is a M ′ ∈ At ( D ) so that ǫ S ( M ) = ǫ S ( M ′ ). Further D ≤ M ′ implies ǫ S ( D ) ≤ ǫ S ( M ′ ) = ǫ S ( M ) ≤ M so that M ∈ At ( ǫ S ( D )).Conversely, if M ∈ At ( ǫ S ( D )) then ǫ S ( D ) ≤ M . We have D ≤ ǫ S ( M ) · D . Weclaim that ǫ S ( M ) · D = 0. Indeed, otherwise we would have ǫ S ( M · ǫ S ( D )) = ǫ S ( M ) · ǫ S ( D ) = ǫ S ( ǫ S ( M ) · D ) = ǫ S (0) = 0 implying M · ǫ S ( D ) = 0 which contradicts M ∈ At ( ǫ S ( D )). So there exists an atom M ′ ∈ At ( ǫ S ( M ) · D ) and thus D ≤ ǫ S ( M ) · D ≤ M ′ . We conclude that M ′ ∈ At ( D ).Further ǫ S ( ǫ S ( M ) · D ) = ǫ S ( M ) · ǫ S ( D ) ≤ ǫ S ( M ′ ), hence ǫ S ( M ) · ǫ S ( M ′ ) · ǫ S ( D ) = ǫ S ( M ′ ). This implies ǫ S ( M ) · ǫ S ( M ′ ) = 0. Since ǫ S ( M ) · ǫ S ( M ′ ) = ǫ S ( M · ǫ S ( M ′ )),we conclude that M · ǫ S ( M ′ ) = 0, hence ǫ S ( M ′ ) ≤ M since M is an atom. It followsthat ǫ S ( M ′ ) ≤ ǫ S ( M ).Proceed in the same way from ǫ S ( M ) · ǫ S ( M ′ ) = ǫ S ( ǫ S ( M ) · M ′ ) in order to ob-tain ǫ S ( M ) ≤ ǫ S ( M ′ ) so that finally ǫ S ( M ) = ǫ S ( M ′ ). But this means that M ∈ σ S ( At ( D )). So we have proved that At ( ǫ S ( D )) = σ S ( At ( D )). ⊓⊔ This means that Φ is embedded in the set algebra P ( At (Φ)) by the map D At ( D ),the algebra of coherent sets of gambles is essentially a set algebra in the technicalsense used here. Note that this is purely a result of the theory of informationalgebra for atomistic algebras, and is not particular to the algebra of coherent setsof gambles. Recall however that we have already seen that Φ under informationorder is a complete lattice. And in fact the map D At ( D ) preserves arbitraryjoins. Corollary 1
Let D j , j ∈ J be an arbitrary family of coherent sets of gambles dom-inated by a coherent set of gambles. Then At ( _ j ∈ J D j ) = \ j ∈ J At ( D j ) . Proof.
The proof of item 1 of Theorem 7 carries over to this more general case:By the assumption W j ∈ J D j = 0. So there is an atom M ∈ At ( W j ∈ J D j ) and from D j ⊆ W j ∈ J D j we conclude that M ∈ At ( D j ), hence M ∈ T j ∈ J At ( D j ). Conversely, ALGEBRAS OF LOWER AND UPPER PREVISIONS M ∈ T j ∈ J At ( D j ), then D j ⊆ M for all j , hence W j ∈ J D j ⊆ M and therefore M ∈ At ( W j ∈ J D j ). ⊓⊔ The set algebra P (Ω) introduced at the beginning of the section is embedded into Φ(as an information algebra and a lattice), hence into P ( At ( φ )). Note that singletonsets { ω } are atoms in the subset algebra P (Ω), and therefore D { ω } are also atoms inthe algebra of coherent sets of gambles, D ω ∈ At (Φ). We may ask how the images At ( D ) of coherent sets of gambles are characterized in At (Φ). A (partial) answer isgiven in Section 8. We remark also that an analogous analysis can be made relativeto the labeled view. We refer to (Kohlas, 2003) for this. In this view, the labeledalgebras Ψ or ˜Ψ are isomorphic to a generalized relational algebra in the sense ofrelational database theory (Kohlas, 2003). We come back to this in Section 8 withregard to lower previsions. Associated with a set of gambles D on L (Ω) is the lower prevision (Troffaes & De Cooman, 2014;Walley, 1991): P¯( f ) = sup { µ ∈ R : f − µ ∈ D } . (7.1)There is also an associated upper prevision ¯ P defined by ¯ P ( f ) = − P¯ ( − f ). Notethat P¯ ( f ) is defined only if the set { µ : f − µ ∈ D } is not empty and bounded fromabove This is the case for all the gambles of a set D contained in a coherent set ofgamble. Since P¯ depends on the set of gambles D , we write P¯ = σ ( D ) and denotethe set of gambles for which P¯ is defined as dom ( σ ( D )). Lemma 7
For a set of gambles D ⊆ L (Ω) we have1. if
6∈ E ( D ) , then D ⊆ dom ( σ ( D )) ,2. if D ∈ C (Ω) , then dom ( σ ( D )) = L (Ω) .Proof. f ∈ D . Then the set { µ : f − µ ∈ D } is not empty, since itcontains at least 0. Assume f − µ ∈ D . If µ ≥ sup f , then f ( ω ) − µ ≤ ω , butthen 0 ∈ E ( D ) which is a contradiction. So, the set { µ : f − µ ∈ D } is not emptyand bounded from above for every f ∈ D , which means that D ⊆ dom ( σ ( D )).2.) If D is a coherent set of gambles, then 0
6∈ C ( D ) = E ( D ) so that D ⊆ dom ( σ ( D ).Consider therefore f ∈ L (Ω) − D . If there would be a µ ≥ f − µ ∈ D ,then f − µ ≤ f ∈ D , which is a contradiction. Now, if µ ≤ inf f <
0, then f − µ ∈ L + (Ω) ∈ D , so it follows inf f ≤ σ ( D )( f ) < dom ( σ ( D )) = L (Ω). ⊓⊔ If D is a coherent set of gambles, then the associated function P¯ on L (Ω) is calleda coherent lower prevision. It is characterized by the following properties: for every f, g ∈ L (Ω), (Walley, 1991), ALGEBRAS OF LOWER AND UPPER PREVISIONS f ) ≥ inf ω ∈ Ω f ( ω ),2. P¯ ( λf ) = λ P¯ ( f ), ∀ λ > f + g ) ≥ P¯ ( f ) + P¯ ( g ).The conjugate value given by:¯ P ( f ) = inf { µ ∈ R : µ − f ∈ D } = − P¯( − f ) . is called upper prevision.It is called coherent, if the associated lower prevision is.Let P (Ω) denote the family of coherent lower previsions. The map σ maps C (Ω)to P (Ω). This map is not one-to-one as different coherent sets of gambles mayinduce the same lower prevision. We may apply the map σ to almost desirable setsof gambles ¯ D and its range is still P (Ω) and we recall that moreover the map σ restricted to almost desirable sets of gambles is one-to-one (Walley, 1991), andP¯ ( f ) = max { µ : f − µ ∈ ¯ D } , ∀ f ∈ L , ¯ D = { f : P¯ ( f ) ≥ } . (7.2)There is also a one-to-one relation between coherent lower previsions P¯ and strictlydesirable sets of gamble D + , so that, if we restrict σ to strictly desirable sets, wehave, (Walley, 1991)P¯ ( f ) = sup { µ : f − µ ∈ D + } , ∀ f ∈ L , D + = { f : P¯( f ) > } ∪ L + (Ω) . Define the maps τ and ¯ τ from coherent lower previsions to strictly desirable sets ofgambles and almost desirable sets of gambles accordingly by τ (P¯ ) = { f : P¯( f ) > } ∪ L + (Ω) , ¯ τ (P¯ ) = { f : P¯ ( f ) ≥ } . Then τ and ¯ τ are the inverses of the map σ restricted to strictly desirable and almostdesirable sets of gambles respectively. The following lemma shows how coherent,strictly desirable and almost desirable sets are linked relative to the coherent lowerprevisions they induce Lemma 8
Let D be a coherent set of gambles. Then D + = τ ( σ ( D )) ⊆ D ⊆ ¯ τ ( σ ( D )) = ¯ D and σ ( D + ) = σ ( D ) = σ ( ¯ D ) . This result follows also from the fact that, in the sup-norm topology of the linear space L (Ω), astrictly desirable set of gambles D + is the relative interior of a coherent set D plus the non-negative,non-zero gambles and any almost desirable set of gambles ¯ D is the relative closure of a coherent set D (Walley, 1991). ALGEBRAS OF LOWER AND UPPER PREVISIONS Proof.
Let P¯ = σ ( D ). Then f ∈ D + means that 0 < P¯ ( f ) = sup { µ : f − µ ∈ D } ,or f ∈ L + (Ω). If f ∈ L + (Ω) then f ∈ D . Otherwise, there is a δ such that0 < δ < P¯ ( f ) and f − δ ∈ D . Therefore f ∈ D and D + ⊆ D . Further, consider f ∈ D . Then we must have P¯ ( f ) = sup { f : f − µ ∈ D } ≥
0, hence f ∈ ¯ D . Thesecond part follows since τ and ¯ τ are the inverse maps of σ on desirable and almostdesirable sets of gambles. ⊓⊔ Recall that strictly desirable sets of gambles form a subalgebra of the informationalgebra of coherent sets of gambles. Then this result establishes a map D D + forany coherent set of gambles to a strictly desirable set. We shall see that this mappreserves combination and extraction. To prove this theorem we need the followinglemma. Lemma 9
Let D be a coherent set of gambles and D + = τ ( σ ( D )) , if f / ∈ L + (Ω) then f ∈ D + if and only if there is a δ > so that f − δ ∈ D .Proof. One part is by definition of strictly desirable gambles: if f ∈ D + and f / ∈ L + (Ω), then there is a δ > f − δ ∈ D + ⊆ D . Conversely, consider f − δ ∈ D for some δ > D + = { f : σ ( D )( f ) > } ∪ L + (Ω). We have σ ( D ) = sup { µ : f − µ ∈ D } . From f − δ ∈ D we deduce that σ ( D )( f ) >
0, hence f ∈ D + . ⊓⊔ The next theorem establishes that this map is a weak homomorphism, weak, becauseit does not apply when D and D are mutually inconsistent, that is if D · D = 0.Before stating this result we need to introduce a partial order relation on lower pre-visions. Indeed, we define P¯ ≤ Q¯ if dom (P¯ ) ⊆ dom (Q¯ ) and P¯ ( f ) ≤ Q¯ ( f ) for all f ∈ dom (P¯ ). This is a partial order on lower previsions ((Troffaes & De Cooman, 2014)).Note that, if we consider the restriction of this partial order relation on lower pre-visions constructed from set D ′ such that 0 / ∈ E ( D ′ ), σ , τ , ¯ τ preserve order. Theorem 8
Let D , D and D be coherent sets of gambles and S ⊆ I .1. If D · D = 0 , then D · D ( D · D ) + = D +1 · D +2 ,2. ǫ S ( D ) ( ǫ s ( D )) + = ǫ S ( D + ) .Proof. For 1. note first that D +1 ⊆ D and D +2 ⊆ D so that D +1 · D +2 = τ ( σ ( D +1 · D +2 )) ⊆ τ ( σ ( D · D )) = ( D · D ) + , Further ( D · D ) + = τ ( σ ( D · D )) = { f : σ ( D · D )( f ) > } ∪ L + (Ω)So, if f ∈ ( D · D ) + , then either f ∈ L + (Ω) or σ ( D · D )( f ) = sup { µ : f − µ ∈ C ( D ∪ D ) } > . (7.3) ALGEBRAS OF LOWER AND UPPER PREVISIONS f ∈ D +1 · D +2 . In the second case there is a δ > f − δ ∈ C ( D ∪ D ). This means that f − δ = h + λ f + λ f , where h ∈ L + (Ω) ∪ { } , f ∈ D , f ∈ D and λ , λ ≥ f = h + ( λ f + δ/
2) + ( λ f + δ/ . We have f ′ = λ f + δ/ ∈ D and f ′ = λ f + δ/ ∈ D . But this, together with λ f = f ′ − δ/ ∈ D if λ > f ′ ∈ L + (Ω), and λ f = f ′ − δ/ ∈ D if λ > f ′ ∈ L + (Ω), shows according to Lemma 9, that f ′ ∈ D +1 and f ′ ∈ D +2 . So, finally, we have f ∈ D +1 · D +2 = C ( D +1 ∪ D +2 ). This proves that( D · D ) + = D +1 · D +2 .To prove 2. note that D + ⊆ D so that( ǫ S ( D )) + = τ ( σ ( ǫ S ( D ))) ⊇ τ ( σ ( ǫ S ( D + ))) = ǫ S ( D + ) . This is valid, because ǫ S ( D + ) is also strictly desirable. Consider then( ǫ s ( D )) + = τ ( σ ( ǫ S ( D ))) = { f : σ ( ǫ S ( D )) > } ∪ L + (Ω) . Here we have σ ( ǫ S ( D ))( f ) = sup { µ : f − µ ∈ C ( D ∩ L S ) } . So, if f ∈ ( ǫ S ( D )) + , then either f ∈ L + (Ω) in which case f ∈ ǫ S ( D + ) or there is a δ > f − δ ∈ C ( D ∩ L S ) = posi {L + (Ω) ∪ ( D ∩ L S ) } . That is f − δ = h + g where h ∈ L + (Ω) ∪ { } and g ∈ D ∩ L S . Then we have f = h + ( g + δ ) and g ′ = g + δ is still S - measurable and g ′ ∈ D . But, given the fact that g = g ′ − δ ∈ D ∩ L S , fromLemma 9, we have g ′ ∈ D + ∩ L S and therefore f ∈ ǫ S ( D + ). Thus we conclude that( ǫ s ( D )) + = ǫ S ( D + ) which concludes the proof. ⊓⊔ Next, we claim that the map σ restricted to coherent or almost desirable sets ofgambles preserves also infima. Here we define inf { P¯ j : j ∈ J } by inf { P¯ j ( f ) : j ∈ J } for all f ∈ L (Ω). Lemma 10
Let D j , j ∈ J be any family of coherent sets. Then we have σ ( \ j ∈ J D j ) = inf { σ ( D j ) } Proof.
Note that the intersection of the coherent sets D j equals a coherent set D .We have σ ( T j ∈ J D j ) = σ ( D ) = P¯ ≤ σ ( D j ) , ∀ j ∈ J . So P¯ is a lower bound ofthe σ ( D j ) , j ∈ J . Let Q¯ be another lower bound of σ ( D j ) , j ∈ J . Then we have τ (Q¯ ) ⊆ D j for all j ∈ J , by definition of τ and the fact that Q¯ is a lower bound of σ ( D j ), hence τ (Q¯ ) ⊆ T j D j = D . But this implies Q¯ ≤ σ ( D ) = P¯. Thus P¯ is indeedthe infimum of the σ ( D j ) for j ∈ J . ⊓⊔ If P¯ ′ is a lower prevision which is dominated by a coherent lower prevision, then itsnatural extension is defined as the infimum of the coherent lower previsions whichdominate it (Walley, 1991), E (P¯ ′ ) = inf { P¯ coherent : P¯ ′ ≤ P¯ } . (7.4) ALGEBRAS OF LOWER AND UPPER PREVISIONS E (P¯ ′ ) is the minimal coherent lower prevision, which dominates P¯ ′ . Now, weprove the key result, that the map σ commutes with natural extension under certainconditions. Theorem 9
Let D ′ be a set of gambles which satisfies the following two conditions.1.
6∈ E ( D ′ ) ,2. for all f ∈ D ′ − L + (Ω) there exists a δ > such that f − δ ∈ D ′ .Then we have σ ( C ( D ′ )) = E ( σ ( D ′ )) . Proof. If D ′ = L + (Ω), then D ′ ∈ C (Ω) and σ ( C ( D ′ )) = E ( σ ( D ′ )) because the lowerprevision associated with L + (Ω) is already coherent. So, assume that D ′ = L + (Ω).We have then E ( D ′ ) = C ( D ′ ) ∈ C (Ω), so that C ( D ′ ) = \ { D : D coherent , D ′ ⊆ D } . Let P¯ ′ = σ ( D ′ ), then σ ( C ( D ′ )) ≥ P¯ ′ and moreover σ ( C ( D ′ )) is coherent, hence σ ( C ( D ′ )) ≥ E (P¯ ′ ), where E (P¯ ′ ) defined by (7.4).Now, consider any coherent lower prevision P¯ so that P¯ ′ ≤ P¯, and τ (P¯ ) the associatedstrictly desirable set of gambles. We claim that D ′ ⊆ τ (P¯ ). Indeed, if f ∈ D ′ , thenP¯ ′ ( f ) ≥
0. If f ∈ L + (Ω), then f ∈ τ (P¯ ), otherwise, if f ∈ D ′ − L + (Ω), then thereis by assumption a δ > f − δ ∈ D ′ , hence 0 < P¯ ′ ( f ) ≤ P¯ ( f ). But thismeans that f ∈ τ (P¯ ). Since a strictly desirable set of gambles is coherent, it follows: σ ( C ( D ′ )) = σ { \ { D : D coherent , D ′ ⊆ D } ≤ σ ( \ { τ (P¯ ) : D ′ ⊆ τ (P¯ ) } )However, thanks to Lemma 10 we have: σ ( \ { τ (P¯ ) : D ′ ⊆ τ (P¯ ) } ) = inf { P¯ coherent : P¯ ′ ≤ P¯ } = E (P¯ ′ )so that σ ( C ( D ′ )) = E ( σ ( D ′ )), concluding the proof. ⊓⊔ From this result on natural extensions in the two formalisms of coherent sets ofgambles and coherent lower previsions, we can now introduce into P (Ω), like inΦ operations of combination and extraction. As in Section 3 consider a family ofvariables X i , i ∈ I with domains Ω i and subsets S ⊆ I of variables with domains Ω S .Let then for two coherent sets of gambles which are not inconsistent, D · D = 0,with P¯ = σ ( D ) and P¯ = σ ( D ),P¯ ′ ( f ) = σ ( D ∪ D )( f ) = sup { µ : f − µ ∈ D ∪ D } = max { P¯ ( f ) , P¯ ( f ) } or P¯ ′ = max { P¯ , P¯ } . We may take the natural extension of E (P¯ ′ ) to define com-bination of two coherent lower previsions P¯ and P¯ . For extraction, we may takethe natural extension of the marginal P¯ S of P¯, defined as the restriction of P¯ to L S .Thus, in summary, we define P¯ · P¯ and e¯ S (P¯ ) by ALGEBRAS OF LOWER AND UPPER PREVISIONS · P¯ ( f ) = E (max { P¯ , P¯ } )( f ), ∀ f ∈ L (Ω), if max { P¯ , P¯ } is dominated by a coherent lower prevision, P¯ · P¯ ( f ) = ∞ for all f ∈ L (Ω)otherwise.2. Extraction: e¯ S (P¯ )( f ) = E (P¯ S )( f ), ∀ f ∈ L (Ω).The following theorem permits to conclude that the set Ψ of coherent lower previsions P (Ω) augmented by σ ( L (Ω)), defined by σ ( L (Ω)( f ) = ∞ for all f ∈ L (Ω), forms adomain-free information algebra under these operations. Theorem 10
Let D +1 , D +2 and D + be strictly desirable set of gambles and S ⊆ I .Then1. σ ( D +1 · D +2 ) = σ ( D +1 ) · σ ( D +2 ) ,2. σ ( L (Ω))( f ) = ∞ , σ ( L + (Ω))( f ) = inf f for all f ∈ L (Ω) ,3. σ ( ǫ S ( D + )) = e¯ S ( σ ( D + )) .Proof. Assume first that D +1 · D +2 = 0 and let P¯ = σ ( D +1 ), P¯ = σ ( D +2 ). Thenthere can be no coherent lower prevision P¯ dominating both P¯ and P¯ . Indeed,otherwise we would have D +1 = τ (P¯ ) ≤ τ (P¯ ) and D +2 = τ (P¯ ) ≤ τ (P¯ ), where τ (P¯ ) is a coherent set of gambles. But this is a contradiction. Therefore, we have σ ( D +1 · D +2 )( f ) = ∞ = σ ( D +1 ) · σ ( D +2 ), for all gambles f .Let then D +1 · D +2 = 0. Then D +1 · D +2 as well as D +1 ∪ D +2 satisfy the condition ofTheorem 9. Therefore, applying this theorem, we have σ ( D +1 · D +2 ) = σ ( C ( D +1 ∪ D +2 )) = E ( σ ( D +1 ∪ D +2 ))= E (max { σ ( D +1 ) , σ ( D +2 ) } ) = σ ( D +1 ) · σ ( D +2 ) . This proves item 1.Item 2 is obvious.For 3. we remark that D + ∩ L S satisfies the condition of Theorem 9. Thus we obtain σ ( ǫ S ( D + )) = σ ( C ( D + ∩ L S )) = E ( σ ( D + ∩ L S )) . Now, σ ( D + ∩ L S ) = sup { µ : f − µ ∈ D + ∩ L S } . But f − µ ∈ D + ∩L S is only possible if f is S -measurable and f − µ ∈ D + . Therefore,we conclude that σ ( D + ∩ L S ) = σ ( D + ) S . Thus, we have indeed σ ( ǫ S ( D + )) = E ( σ ( D + ) S ) = e¯ S ( σ ( D + )). This concludes the proof ⊓⊔ The map σ , restricted to the information algebra Φ + , is bijective. It follows that theset Ψ = P (Ω) ∪ { σ ( L (Ω) } is a domain-free information algebra, isomorphic to Φ + .There is obviously the connected (isomorphic) information algebra of upper previ-sions. The following corollary, shows furthermore that Φ is weakly homomorphic toΨ. ALGEBRAS OF LOWER AND UPPER PREVISIONS Corollary 2
Let D , D and D be coherent sets of gambles so that D · D = 0 and S ⊆ I . Then1. σ ( D · D ) = σ ( D ) · σ ( D ) ,2. σ ( ǫ S ( D )) = e¯ S ( σ ( D )) .Proof. These claims are immediate consequences of Theorems 8 and 10. ⊓⊔ The homomorphism does not extend to a pair of inconsistent coherent sets of gam-bles, as the following example shows.
Example C: onsider a set of possibilities Ω = { ω , ω } and let D = { f : f ( ω ) > − f ( ω ) } ∪ { f : f ( ω ) = − f ( ω ) , f ( ω ) ≥ , f = 0 } ,D = { f : f ( ω ) > − f ( ω ) } ∪ { f : f ( ω ) = − f ( ω ) , f ( ω ) < } . These are coherent sets of gambles, but they are mutually inconsistent, since wehave D · D = L (Ω) because 0 ∈ E ( D ∪ D ). But on the other hand, D +1 = D +2 = { f : f ( ω ) > − f ( ω ) } and therefore D +1 · D +2 = D +1 = L (Ω). So already the map D D + does not respectinconsistencies, and then σ ( D · D ) = σ ( D ) · σ ( D ) in this example. ⊖ Finally, it follows from Lemma 10 and Corollary 2 that for any family of coherentsets of gambles D j , σ ( ǫ S ( \ j D j )) = e¯ S ( σ ( \ j D j )) = e¯ S (inf { σ ( D j ) } )and σ ( \ j ǫ S ( D j )) = inf { σ ( ǫ S ( D j )) } = inf { e¯ S ( σ ( D j )) } . Therefore it follows from (3.2) also that for any family P¯ j of coherent lower previ-sions, we have e¯ S (inf { P¯ j } ) = inf { e¯ S (P¯ j ) } . So, in the information algebra of lower prevision extraction distributes still overmeet (infimum).As always, there is also a labeled version of this domain-free algebra of lower previ-sion, which will be described very briefly. For any subset S of I let˜Ψ S = { P¯ S : P¯ coherent lower prevision } , where again P¯ S is the restriction of P¯ to L S . Further let˜Ψ = [ S ⊆ I ˜Ψ S . Within ˜Ψ we define the following operations, given P¯ S , P¯ ,S , P¯ ,T ∈ ˜Ψ: LINEAR PREVISIONS d (P¯ S ) = S ,2. Combination: P¯ ,S · P¯ ,T = (P¯ · P¯ ) S ∪ T , where on the right combination oflower previsions is used,3. Projection (Marginalization): for T ⊆ S , π T (P¯ S ) = P¯ T .It is easy to verify that ˜Ψ with these operations satisfy the axioms of a labeledinformation algebra as presented in Theorem 3 for coherent sets of desirable gambles.We remark also that, if we restrict τ to L S , then τ (P¯ S ) = { f : P¯ S ( f ) > } ∪ L + S =( { f : P¯ ( f ) > } ∪ L + (Ω)) ∩ L S = τ (P¯ ) ∩ L S . This relates the labeled algebra ofcoherent lower previsions to the second version of the labeled version of coherent orrather strictly desirable sets of gambles at the end of Section 4 as isomorphic.The results of this section shows that the coherent lower (and upper) previsionsform an information algebra closely related to the information algebras of coherentor desirable sets of gambles. This relationship carries over to the labeled versions ofthe algebras involved, a subject we do not pursue here. However we have seen thatthe algebra of coherent sets of gambles is completely atomistic. In the next sectionwe discuss what this means for the algebra of lower previsions. If P¯ ( f ) = − P¯ ( − f ) for all f in L (Ω), that is if lower and upper prevision coincide,P¯ is called a linear prevision. Then its usual to write P¯ = ¯ P = P . Linear previsionshave an important role in the theory of imprecise probabilities. Therefore, in thissection they will be examined from the point of view of information algebras. Firstof all a linear prevision is a lower (and upper) prevision. So, if P (Ω) denote theset of linear previsions on L (Ω), we have P (Ω) ⊆ P (Ω). Note that from the thirdcoherence property of lower previsions it follows that P ( f + g ) = P ( f ) + P ( g ), forevery f, g ∈ L (Ω).Let us concentrate ourselves on the strictly desirable set of gambles associated witha linear prevision, τ ( P ) = { f : P ( f ) > } ∪ L + (Ω) = { f : − P ( − f ) > } ∪ L + (Ω) . We call these sets maximal strictly desirable sets of gambles. It is possible to showthat we have M + = τ ( P ) = τ ( σ ( M )), where M is a maximal set of gambles, see(Walley, 1991). Since these sets are atoms in the algebra of coherent sets of gambles,we may presume that maximal strictly desirable sets of gambles are atoms in thealgebra of strictly desirable sets of gambles and linear previsions are also atoms inthe algebra of coherent prevision. And this is indeed the case. Lemma 11
Let P¯ be an element of Ψ and P a linear prevision. Then P ≤ P¯ implieseither P¯ = P or P¯ ( f ) = ∞ for all f ∈ L (Ω) . LINEAR PREVISIONS Proof.
Clearly P¯ ( f ) = + ∞ for all f ∈ L is a possible solution. Consider instead thecase in which P¯ is coherent.From (Walley, 1991), we know P¯ ( f ) ≤ ¯ P ( f ) , ∀ f ∈ L (Ω).Then, we have:P¯ ( f ) ≤ ¯ P ( f ) = − P¯ ( − f ) ≤ − P ( − f ) = P ( f ) , ∀ f ∈ L (Ω) . (8.1)Given the fact that, by hypothesis, we have also P¯ ( f ) ≥ P ( f ) , ∀ f ∈ L (Ω), we havethe result. ⊓⊔ Lemma 12
Let D + be an element of Φ + and M + a maximal strictly desirable setof gambles. Then M + ≤ D + implies either D + = M + or D + = L (Ω) .Proof. M + ≤ D + ⇒ σ ( M + ) ≤ σ ( D + ) . (8.2)From Lemma 11, we have σ ( D + )( f ) = + ∞ , ∀ f ∈ L (Ω) or σ ( D + ) = σ ( M + ). Inthe first case, we have D + = L (Ω). In the second one, we have D + = τ ( σ ( D + )) = τ ( σ ( M + )) = M + . ⊓⊔ From Lemma 11, we may automatically deduce the properties of atoms in an infor-mation algebra, so, for example, we have P¯ · P = P or P¯ · P = 0, where here 0 isthe null element P¯ ( f ) = ∞ , ∀ f ∈ L (Ω). The information algebra Φ of coherent setsis completely atomistic. It is to be expected that the same holds for the algebra Ψof coherent lower previsions. Let At (Ψ) = P (Ω) be the set of all linear previsions(atoms) and At (P¯ ) the set of all linear previsions (atoms) dominating the coherentlower prevision P¯, At (P¯ ) = { P ∈ At (Ψ) : P¯ ≤ P } . Then the following theorem shows that the information algebra Ψ is completelyatomistic.
Theorem 11
In the information algebra of lower previsions Ψ , the following holds:1. If P¯ is a coherent lower prevision, thenP¯ = inf At ( P¯ ) .
2. If A is any non-empty subset of linear prevision in At (Ψ) , thenP¯ = inf A exists and is a coherent lower prevision. For the proof of this theorem, see (Walley, 1991, Theorem 3.3.3).
LINEAR PREVISIONS A is any non-empty family of linear previsions on L (Ω),then inf A exists and is a coherent lower prevision P¯ . Then we have A ⊆ At (P¯ ) andit follows P¯ = inf A = inf At (P¯ ) . So, the coherent lower prevision P¯ is the lower envelope of the linear previsions(atoms) which dominate it.As any atomistic information algebra, Ψ is embedded in the set algebra of subsets of At (Ψ) by the map P¯ At (P¯ ). This rises the question how to characterize the imagesof Ψ in the algebra of subsets At (Ψ). The answer is given by the weak* compactnesstheorem (Walley, 1991): The sets At (P¯ ) for any coherent lower prevision P¯ areexactly the weak* compact convex subsets of At (Ψ) in the weak* topology on At (Ψ).There are many other sets of linear previsions A whose lower envelope equals P¯ . IfP¯ = inf A and A ⊆ B ⊆ At (P¯ ), then P¯ = inf B . In fact there is a minimal set E ⊆ At (P¯ ) so that P¯ = inf E and this is the set of extremal points in the convex set At (P¯ ). This follows from the extreme point theorem (Walley, 1991).We shall come back to the embedding of the algebra of coherent lower previsions inthe algebra of subsets of At (Ψ) below in the labeled view of the algebra.We now examine linear previsions in the light of extraction or, equivalently, in thelabeled view of the information algebra of coherent lower previsions, see Section 7.Recall that for any subset S of I , P¯ S denotes the restriction of the coherent lowerprevision P¯ to L S . Similarly, P S is the restriction of the linear prevision to L S .These element P S are local atoms in the labeled information algebra ˜Ψ, that is, if P S ≤ P¯ S , then either P S = P¯ S or P¯ S = 0 S for any linear prevision P and P¯ ∈ ˜Ψ andwhere 0 S is the null element for label S , that is 0 S ( f ) = ∞ for all f ∈ L S . We call P S also a locally linear previsions. There follow as usual some elementary propertiesof linear previsions, as atoms, for instance such as the equivalent of Lemma 3. Lemma 13
Assume P and P , P to be linear previsions, that is atoms and furtherP¯ ∈ ˜Ψ . Consider also S ⊆ I . Then • P S · P¯ S = 0 S or = P S , • if T ⊆ S , then π T ( P S ) is locally linear relative to T , that is an atom relativeto T , • either P¯ S ≤ P S or P¯ S · P S = 0 S , • P ,S · P ,S = 0 S or P ,S = P ,S . For a proof of this result on atoms in labeled linear algebras we refer to (Kohlas, 2003).Just as the labeled information algebra of coherent sets of gambles is atomic, atom-istic and completely atomistic, the same holds for the labeled algebra of coherent
LINEAR PREVISIONS At ( ˜Ψ S ) be the set of all locally linear previsions relative to S and At (P¯ S ) the set of locally linear previsions, or atoms, dominating the coherentlower prevision P¯ S . • Atomic:
For every coherent lower prevision P¯ there is a linear prevision P sothat P¯ S ≤ P S . That is, At (P¯ S ) is not empty. • Atomisitic : For every coherent lower previsions P¯ we have P¯ S = inf At (P¯ S ). • Completely Atomistic:
For any, not empty, subset A of At ( ˜Ψ S ), inf A existsand belongs to ˜Ψ S .It is further well-known that the local atoms of any atomic labeled informationalgebra satisfy the following conditions, expressed here for locally linear previsions(Kohlas, 2003). Lemma 14
Let At ( ˜Ψ) = [ S ⊆ I At ( ˜Ψ S ) and let P S , P T , P ,S , P ,T ∈ At ( ˜Ψ) .Then,1. if T ⊆ d ( P S ) , then d ( π T ( P S )) = T ,2. if T ⊆ R ⊆ d ( P S ) , then π T ( π R ( P S )) = π T ( P S ) ,3. π S ( P S ) = P S ,4. if π S ∩ T ( P ,S ) = π S ∩ T ( P ,T ) , then there is a P S ∪ T ∈ At ( ˜Ψ) so that π S ( P S ∪ T ) = P ,S and π T ( P S ∪ T ) = P ,T ,5. for an element P S , if S ⊆ T , there is a P T ∈ At ( ˜Ψ) so that π S ( P T ) = P S . Most of these properties are immediate consequences from the lower previsions beinga labeled information algebra. For a proof of item 4 we refer to (Kohlas, 2003). Asystem like At ( ˜Ψ) with a labeling and a projection operation satisfying the conditionsof Lemma 14 is called a tuple system , since it abstracts the properties of concretetuples as used in relational database systems. And generalized relational algebrascan be defined using tuple systems and they turn out to be labeled informationalgebras (Kohlas, 2003). In the case of the tuple system At ( ˜Ψ) this goes as follows:A subset R of At ( ˜Ψ S ) is called a (generalized) relation on S . Denote by R S all theserelations on S and let R = [ S ⊆ I R S . Then, in R , we define the following operations. LINEAR PREVISIONS
Labeling: d ( R ) = S if R ∈ R S .2. Natural Join: R ⊲⊳ R = { P S ∪ T ∈ At ( ˜Ψ S ∪ T ) : π S ( P S ∪ T ) ∈ R , π T ( P S ∪ T ) ∈ R } if d ( R ) = S and d ( R ) = T .3. Projection: π T ( R ) = { π T ( P S ) : P S ∈ R } , if d ( R ) = S .With these operations, natural join as combination, the algebra of relations R be-comes a labeled information algebra. Note that the null element of natural join isthe empty set and the unit with label S is At ( ˜Ψ S ). This depends only on At ( ˜Ψ)being a tuple system (Kohlas, 2003). What is more, it turns out that the atom-istic labeled algebra of coherent lower previsions is embedded into this generalizedrelational algebra. Theorem 12
Let ˜Ψ be the labeled information algebra of coherent lower previsions.let further P¯ and Q¯ be elements of ˜Ψ . Then1. At ( P¯ S · Q¯ T ) = At ( P¯ S ) ⊲⊳ At ( Q¯ T ) ,2. At ( π T ( P S ) = π T ( At ( P S )) if T ⊆ S . Again, this is a theorem of atomistic information algebras and not particular tolower previsions, (Kohlas, 2003). It says that the map P¯ S At (P¯ S ), complementedby 0 S
7→ ∅ , is a homomorphism between the labeled information algebra of coher-ent lower previsions and the generalized relational algebra of sets of locally linearprevisions. Furthermore, since the map is obvioulsy one-to-one, it tells us that ˜Ψ isembedded into this relational algebra. This is the labeled version of the embeddingof the domain-free algebra into a set algebra of atoms.By definition, any linear prevision is also locally linear relative to any subset ofvariables. Fix any linear prevision Q on L S and S, T ⊆ I . Then M ( Q ) = { P ∈ P : P S = Q } is called the linear extension of Q to L (Ω). Now, define Q¯ = E ( Q ) to be thenatural extension of Q to L (Ω). Then, by the definition of extraction we havee¯ s (P¯ ) = E (P¯ S ) = Q¯ whenever P¯ S = Q and P¯ is coherent. In particular, we havee¯ S ( P ) = Q¯ for all P in the linear extension of Q , then P ≥ e¯ S ( P ) = Q¯ , hence P ∈ At (Q¯ ). Conversely, if P ∈ At (Q¯ ), then P S ≥ Q , but this is only possible if P S = Q . It follows that M ( Q ) = At (Q¯ ). This is a version of the natural extensiontheorem (Walley, 1991). Since extraction distributes over meet, we haveQ¯ = e¯ S (Q¯ ) = e¯ S (inf At (Q¯ )) = inf e¯ S ( At (Q¯ ))and also Q¯ = e¯ S (Q¯ ) = e¯ S (inf M ( Q )) = inf e¯ S ( M ( Q ))If P ∈ M ( Q ) = At (Q¯ ), then e¯ S ( P ) = Q¯ , hence P ≥ Q¯ implies e¯ S ( P ) = Q¯ . THE MARGINAL PROBLEM Consistency, inconsistency, compatibility or incompatibility, whatever is exactlymeant by these concepts, are general properties of information. Here these no-tions will be defined and studied with respect to coherent sets of gambles, but in theframe and using results of information algebras. Two pieces of information can beconsidered as compatible, if they are not contradictory, that is, if their combinationis not the null element. Hence a finite family of coherent sets of gambles D , . . . , D n is compatible, if 0 D · . . . · D n = E ( ∪ ni =1 D i ). This is called “avoiding partial loss”in desirability (Miranda & Zaffalon, 2020). Otherwise the family is called incompat-ible. There is, however, a more restrictive concept of compatibility. Here a familyof coherent sets of gambles D , . . . , D n , where D i has support S i , i = 1 , . . . , n , iscalled compatible, if there is a coherent set of gambles D such that ǫ S i ( D ) = D i for i = 1 , . . . , n , see (Miranda & Zaffalon, 2020). To decide whether a family of D i iscompatible in this sense is also called the marginal problem, since extractions are (inthe labeled view) projections or marginals. In the view of information algebra, weprefer to call this type of compatibility consistency , since the pieces of information D i come from or are part of the same piece of information D .In (Miranda & Zaffalon, 2020) two coherent sets D i and D j , where D i has support S i and D j support S j , are called pairwise compatible, or, in our terminology pairwiseconsistent , if D i ∩ L S j = D j ∩ L S i or E S j ( D i ) = E S i ( D j ) . In terms of the information algebra this means that ǫ S j ( D i ) = C ( E S j ( D i )) = C ( E S i ( D j )) = ǫ S i ( D j ) (9.1)From this it follows that ǫ S i ∩ S j ( D i ) = ǫ S i ∩ S j ( ǫ S j ( D i )) = ǫ S i ∩ S j ( ǫ S i ( D j )) = ǫ S i ∩ S j ( D j ) . In an information algebra in general we could take this as a definition of pairwiseconsistency. From this we may recover (9.1), since by item 5 of the list of propertiesof support (Section 4), if S i is a support of D i and S j of D j , we have ǫ S j ( D i ) = ǫ S i ∩ S j ( D i ) and ǫ S i ( D j ) = ǫ S i ∩ S j ( D j ).Now let D = D i · D j and D i and D j pairwise consistent. Then ǫ S i ( D ) = D i · ǫ S i ( D j ) = D i · ǫ S j ( D i ) = D i and also ǫ S j ( D ) = D j . So, pairwise consistent piecesof information are consistent. And, conversely, if D i and D j are consistent, then ǫ S i ∩ S j ( D ) = ǫ S i ∩ S j ( ǫ S i ( D )) = ǫ S i ∩ S j ( D i ) and similarly ǫ S i ∩ S j ( D ) = ǫ S i ∩ S j ( D j ) andthe two elements are pairwise consistent.It is well-known that pairwise consistency among a family of D , . . . , D n of piecesof information is not sufficient for the family to be consistent. It is also well-knownthat a sufficient condition to obtain consistency from pairwise consistency is that thefamily of supports S , . . . , S n of the D i satisfy the the running intersection property(RIP, that is form a join tree or a hypertree construction sequence): THE MARGINAL PROBLEM RIP
For i = 1 to n − p ( i ), i + 1 ≤ p ( i ) ≤ n such that S i ∩ S p ( i ) = S i ∩ ( ∪ nj = i +1 S j ) . Then we have the following theorem, see prop. 1 and theorem 2 in (Miranda & Zaffalon, 2020),a theorem, which in fact is a theorem of information algebras in general.
Theorem 13 If D , . . . , D n , a family of compatible coherent sets of gambles, withsupports S , . . . , S n which satisfy RIP, are pairwise consistent, then they are consis-tent and ǫ S i ( D · . . . · D n ) = D i for i = 1 , . . . , n .Proof. We give a proof in the framework of general information algebras. Let Y i = S i +1 ∪ . . . ∪ S n for i = 1 , . . . , n − D = D · . . . · D n . Then by RIP ǫ Y ( D ) = ǫ Y ( D ) · D · . . . · D n = ǫ Y ( ǫ S ( D )) · D · . . . · D n = ǫ S ∩ Y ( D ) · D · . . . · D n = ǫ S ∩ S p (1) ( D ) · D · . . . · D n . But by pairwise compatibility ǫ S ∩ S p (1) ( D ) = ǫ S ∩ S p (1) ( D p (1) ), hence by idempo-tency ǫ Y ( D ) = D · . . . · D n . By induction on i , one shows exactly in the same way that ǫ Y i ( D ) = D i +1 · . . . · D n , ∀ i = 1 , .., n − . So, we obtain ǫ S n ( D ) = ǫ Y n − ( D ) = D n . Now, we claim that ǫ S i ( D ) = ǫ S i ∩ S p ( i ) ( D ) · D i . Since S p ( i ) ⊆ Y i , we have by RIP D i · ǫ S i ∩ S p ( i ) ( D ) = D i · ǫ S i ∩ S p ( i ) ( ǫ Y i ( D )) = D i · ǫ S i ∩ Y i ( ǫ Y i ( D )) = D i · ǫ S i ( ǫ Y i ( D ))= D i · ǫ S i ( D i +1 · . . . · D n ) = ǫ S i ( D i · D i +1 · . . . · D n ) = ǫ S i ( ǫ Y i − ( D ))= ǫ S i ( D ) . Then, by backward induction, based on the induction assumption ǫ S j ( D ) = D j for j > i , and rooted in ǫ S n ( D ) = D n , for i = n − , . . .
1, we have by pairwise consistency ǫ S i ( D ) = ǫ S i ∩ S p ( i ) ( D ) · D i = ǫ S i ∩ S p ( i ) ( ǫ S p ( i ) ( D )) · D i = ǫ S i ∩ S p ( i ) ( D p ( i ) ) · D i = ǫ S i ∩ S p ( i ) ( D i ) · D i = D i . This concludes the proof. ⊓⊔ Note that the condition ǫ S i ( D · . . . · D n ) = D i implies that the family D , . . . , D n is consistent. So, this is a sufficient condition for consistency. This theorem is atheorem of information algebra, it holds not only for coherent sets of gambles, butfor any information algebra, in particular for the algebra of coherent lower previsionsfor instance.The definition of consistency and pairwise consistency depend on the supports (henceindirectly on the S -measurability) of the elements D i . But D i may have different THE MARGINAL PROBLEM D i and D j are pairwiseconsistent according to their supports S i and S j , that is ǫ S i ∩ S j ( D i ) = ǫ S i ∩ S j ( D j ). Itmay be that a set S ′ i ⊆ S i is still a support of D i and a subset S ′ j ⊆ S j a support of D j . Then ǫ S ′ i ∩ S ′ j ( D i ) = ǫ S ′ i ∩ S ′ j ( ǫ S i ∩ S j ( D i )) = ǫ S ′ i ∩ S ′ j ( ǫ S i ∩ S j ( D j )) = ǫ S ′ i ∩ S ′ j ( D j ) . So, D i and D j are also pairwise consistent relative to the smaller supports S ′ i and S ′ j . The finite supports of a coherent set of gambles D i , have a least support d i = \ { S : S support of D i } . This is called the dimension of D i , it is itself a support of D i (see item 4 on the listof properties of supports). So, if D i and D j are pairwise consistent relative to twoof their respective supports S i and S j they are pairwise consistent relative to theirdimensions d i and d j . This makes pairwise consistency independent of an ad hocselection of supports.But what about consistency? Assume that the family D , . . . , D n is consistent rel-ative to the supports S i of D i , that is ǫ S i ( D ) = D i . Then we have ǫ d i ( D ) = ǫ d i ( ǫ S i ( D )) = ǫ d i ( D i ) = D i . So, the family D , . . . , D n is also consistent with respect to the system of their di-mensions. Again, this makes the definition of consistency independent of a particularselection of supports. We remark that the set { S : S support of D i } is an upset,that is with any element S in the set an element S ′ ⊇ S belongs also to the set (item6 on the list of properties of supports). In fact, { S : S support of D i } = ↑ d i is the set of all supersets of the dimension. Now, assume D , . . . , D n pairwise con-sistent. The dimensions d i may not satisfy RIP, but some sets S i ⊇ d i may (forminga covering join tree for the family D , . . . , D n ). Then by Theorem 13 and thisdiscussion, D , . . . , D n are consistent.From a point of view of information, consistency of pieces D , . . . , D n of informationis not always desirable. It is a kind of irrelevance or (conditional) independencecondition. In fact, if the members of the family D , . . . , D n are pairwise consistent,and the supports S i satisfy RIP, then D i = ǫ S i ( D · . . . · D n ) means that, giventhe information on the separators S i ∩ S j , the pieces of information D j for j = i give no new information relative to variables in S i . If, on the other hand, thefamily of pieces of information D , . . . , D n is not consistent, but compatible in thesense that D = D · . . . · D n = 0, then, if S to S n satisfy RIP, we have that thefamily ǫ S i ( D ) ≥ D i (in the information order), that is D j may furnish additionalinformation on the variables in S i for i = j .in (Kohlas, 2003), we have D = ǫ S ( D ) · . . . · ǫ S n ( D ) THE MARGINAL PROBLEM D ′ i = ǫ S i ( D ) are pairwise consistent and by definition consistent (thisis remark 1 in (Miranda & Zaffalon, 2020)).To conclude, note that most of this discussion of consistency (in particular Theorem13) depends strongly on idempotency E2 of the information algebra. For instancethe valuation algebra corresponding to Bayesian networks is not idempotent, aswell as many other semiring-valuation algebras (Kohlas & Wilson, 2006). The RIPcondition, Theorem 13), does not apply in these cases.We have remarked that consistency is essentially an issue of information algebra.So, we may expect that the concepts and results on consistency of coherent sets ofgambles carry over to coherent lower and upper previsions. First of all we definethe concept of support , also for coherent lower previsions. Analogously to coherentsets of gambles, we say that a subset S of I is called a support of a coherent lowerprevision P¯ , if there exists a lower prevision Q¯ on L S such that P¯ is the naturalextension of Q¯ , that is P¯ = E (Q¯ ).Coherent lower previsions P¯ , . . . , P¯ n with supports S , . . . , S n are called consistent,if there is a coherent lower prevision P¯ such that e¯ S i (P¯ ) = P¯ i for all i = 1 , . . . , n .Recall that if P¯ i has support S i , there is a lower prevision Q¯ i on L S i such that P¯ i is the natural extension of Q¯ i , that is P¯ i = E (Q¯ i ), hence also P¯ i | S i = Q¯ i . We mayalso call the lower previsions Q¯ i consistent, if there is a lower prevision P¯ such thate¯ S i (P¯ ) = E (Q¯ i ). If P¯ , . . . , P¯ n are consistent, then all pairs P¯ i and P¯ j are pairwiseconsistent, that is e¯ S i ∩ S j (P¯ j ) = e¯ S i ∩ S j (P¯ i ). This implies also e¯ S j (P¯ i ) = e¯ S i (P¯ j ) orP¯ i | S i ∩ S j = P¯ j | S i ∩ S j .Theorem 13 carries over, since it is in fact a theorem of information algebras. Theorem 14
If P¯ , . . . , P¯ n , compatible coherent lower previsions with supports S , . . . , S n which satisfy RIP, are pairwise consistent, then they are consistent and e¯ S i ( P¯ · . . . · P¯ n ) = P¯ i for i = 1 , . . . , n . Of course, there are close relations between consistency of coherent sets of gamblesand lower previsions. If D , . . . , D n are consistent coherent sets of gambles withsupports S , . . . , S n , then the associated lower previsions σ ( D i ) are consistent too,since e¯ S i ( σ ( D )) = σ ( ǫ S i ( D )) = σ ( D i ). Conversely, if P¯ , . . . , P¯ n are consistentlower previsions with support S to S n , then there is a family of consistent strictlydesirable sets of gambles D + i = τ (P¯ i ) with P¯ i = σ ( D + i ). More generally, if P¯ is thelower prevision so that e¯ S i (P¯ ) = P¯ i , and P¯ = σ ( D ), then [ ǫ S i ( D )] σ = [ D i ] σ , where σ ( D i ) = P¯ i and [ D ] σ is the equivalence class of coherent sets of gambles associatedwith the homomorphism σ , that is D ≡ σ D ′ if σ ( D ) = σ ( D ′ ). We refer to thehomomorphism theorem of universal algebra, see for instance (Kohlas, 2003). Soconsistency among coherent sets of gambles is a class property.Let’s turn to coherent locally linear previsions P¯ , . . . , P¯ n relative to S to S n . Recallthat then P¯ i | S i = Q i is a linear prevision on L S i , ∀ i = 1 , ..., n . We claim that if thesequence S , . . . , S n satisfies RIP, and the the elements of P¯ , . . . , P¯ n are pairwiseconsistent, then the whole family is consistent. This is expressed in the followingtheorem. Theorem 15
If P¯ , . . . , P¯ n are compatible locally linear previsions relative to S to S n , pairwise consistent and if S , . . . , S n satisfies RIP, then P¯ , . . . , P¯ n are consistentand there is a linear prevision P so that ǫ S i ( P ) = P¯ i for all i = 1 , . . . , n .Proof. Let P¯ = P¯ · . . . · P¯ n . By Theorem 14 this is a coherent lower previsionand ǫ S i (P¯ ) = P¯ i . Then there exists a linear prevision (an atom) P ∈ At (P¯ ) sincethe algebra of coherent lower previsions is atomic. We have then P¯ ≤ P , henceP¯ i = ǫ S i (P¯ ) ≤ ǫ S i ( P ). But ǫ S i ( P ) and P¯ i are both local atoms (locally linear),therefore we must have P¯ i = ǫ S i ( P ). ⊓⊔ This last theorem is again a theorem of information algebra. Therefore, it appliesequally to the atomic algebra of coherent sets of gambles. But linear previsionshave a special appeal since they are linked to probabilities, that is why we chooseto present the theorem in this frame.
10 Outlook
This paper presents a first approach to information algebras related to desirablegambles, lower and upper previsions. There are many aspects and issues whichare not addressed here. Foremost is the issue of conditioning. Conditioning of acoherent set of gambles on a event or set B could be seen as combination of thecoherent set with B . This concept of conditioning however has yet to be developedand compared with the usual approach to conditioning in imprecise probabilities asproposed originally in (Walley, 1991). Further, in the multivariate model consideredin this paper, desirable gambles and previsions are considered relative to linear spaces L (Ω) and L S of gambles. In the view of information algebra then coherent sets of S -measurable gambles represent pieces of information or belief relative to sets S ofvariables. More general families of spaces of gambles may be considered, for instancefamilies of gambles measurable relative to certain partitions of the set of possibilitiesΩ or certain lattices of Borel fields, etc. How can the concept of information algebrasbe adapted to such more general frames? Both of these subjects would also serveto deepen the issue of conditional independence, which seems to be fundamental forany theory of information. References
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