revision rules for convex sets of probabilities introduction - CiteSeerX

REVISION RULES FOR CONVEX SETS OF PROBABILITIES Serafn Moral1, and Nic Wilson2 1Departamento

de Ciencias de la Computacion Universidad de Granada 18071 - Granada - Spain 2Department of Computer Science Queen Mary and West eld College Mile End Rd., London E1 4NS, UK

INTRODUCTION The best understood and most highly developed theory of uncertainty is Bayesian probability. There is a large literature on its foundations and there are many dierent justi cations of the theory; however, all of these assume that for any proposition a, the beliefs in a and :a are strongly tied together. Without compelling justi cation, this assumption greatly restricts the type of information that can be satisfactorily represented, e.g., it makes it impossible to represent adequately partial information about an unknown chance distribution P such as 0:6  P(a)  0:8. The strict Bayesian requirement that an epistemic state be a single probability function seems unreasonable. A natural extension of the Bayesian theory is thus to allow sets of probability functions and to consider constraints and bounds on these, and to calculate supremum and in mum values of the probabilities of propositions (known as Upper and Lower Probabilities) given the constraints. Early work on this includes Boole1 and Good2 and early appearances in the Arti cial Intelligence literature include Quinlan3 and Nilsson4. Walley's Imprecise Probability5 is arguably the most satisfactory of all current theories of uncertain reasoning from a foundational point of view. It is a behavioural version of Upper and Lower Probability built on the work of Smith6 and Williams7. Mathematically, it is similar to other Lower Probability theories but it is based on gambles, with coherence axioms relating the gambles that an agent nds desirable; from here it is a small step to turning the system into a logic. In next section, we give the de nition of this logic with a language, very simple proof theory and semantics. The results of this section can be found in Wilson and Moral8.

In the rest of the paper we study the problem of belief change. A belief change occurs when we add new information to a knowledge state. In convex sets of probabilities, there are three main ways of belief change. The rst is expansion. On it we have new general information (rules). We simply add these new rules. If there is inconsistency it is not resolved and everything can be obtained from the new belief state. Revision is dierent from expansion on the handling of inconsistency9. When an inconsistency between the new and old knowledge appears, then part of the old knowledge is retracted to achieve consistency. Finally focussing is when we add new information consisting in observations for a particular case (facts)10;11. The distinction between revision and focusing is not always clear. This is mainly due to the fact that in classical probability, revision and focusing are carried out with the same procedure: conditioning. However, in convex sets of probabilities the dierences between these two belief change operations are evident, given rise to dierent procedures. The main contributions of the paper are the expression of these procedures in the logical language about probabilities and the extensive study of revision procedures. Some known rules to operate with probability bounds turn to be particular cases of revision or focusing. An interesting relationship between focusing and revision of convex sets of probabilities is established, by considering possibility distributions on convex sets of probabilities.

THE LOGIC OF GAMBLES Let be a nite set of possibilities, exactly one of which must be true. A gamble on is a function from to IR. If you were to accept gamble X and ! turned out to be true then you would gain X (!) utiles (so you would lose if X (!) < 0). An agent's beliefs are elicited by asking them to tell us a set ? of gambles they nd acceptable, i.e., gambles they would be happy to accept. For  2 IR we write  for the constant gamble de ned by (!) =  for all ! 2 . If  > 0 we should certainly consider gamble  acceptable since, whatever happens, we gain. If  < 0 then we should certainly not accept gamble  since, whatever happens, we lose. Addition and subtraction of gambles is de ned pointwise, so that for gambles X and Y , for each ! 2 , (X ? Y )(!) = X (!) ? Y (!). For a  we de ne gamble a to be the indicator function of a, so that a(!) = 1 if ! 2 a and a(!) = 0 otherwise. Any such gamble a would seem acceptable since you couldn't lose. On the other hand, the gamble a ? 1 (given by (a ? 1)(!) = 0 if ! 2 a; (a ? 1)(!) = ?1 otherwise) would only be acceptable to you if you were certain that a were true. If = f!1; !2g and X is the gamble f!1g ? 0:5 (so that X (!1 ) = 0:5 and X (!2 ) = ?0:5) and you considered !1 more likely than !2 then it seems that you should consider the gamble X acceptable.

The Expressiveness of Gambles Gambles can represent a surprisingly varied set of probability statements. Another way of viewing gambles is as linear constraints on an unknown chance distribution, and indeed any such linear inequality can be represented as a gamble. For example, the constraint Pr(a)  0:6 would be represented by the gamble a ? 0:6; the constraint Pr(a)  Pr(b) + 0:1 would be represented by the gamble b + 0:1 ? a; the constraint Pr(ajb)  0:9 by the gamble 0:9b ? (a \ b). In general a linear constraint 1 Pr(a1) +


+ k Pr(ak )   is represented by the gamble X = 1a1 +


+ k ak ? . The constraint 1 Pr(a1)+


+ k Pr(ak )   is equivalent to the constraint (?1) Pr(a1)+




+ (?k ) Pr(ak )  ? and so is represented by the gamble ?X . Hence 1 Pr(a1) +


+ k Pr(ak ) =  is represented by the pair of gambles fX; ?X g. The language

does, however, have some limitations: independence relationships or constraints such as Pr(ajb)  Pr(a) cannot be represented simply in terms of gambles, since they are non-linear.

The Language Let L be the set of gambles on . The constant gamble 1 we write as > and the constant gamble ?1 we write as ?. Proof Theory We are going to de nei an inference relation `, where for ?  L and X 2 L, ? ` X is intended to mean `Given that I'm prepared to accept any gamble in ? then I should also be prepared to accept X '. The axioms and inference rules used to de ne ` will be justi ed by the semantics (with probability functions as models) given later. However, as Walley shows, they can be justi ed directly (and arguably more satisfactorily, since then the theory is not based on additive probability functions). Axiom Schema: X , for any X with minX  0. For any such X we can't lose so it should be acceptable. Inference Rule (Schema) 1: For any  2 IR such that  > 0 the inference rule From X deduce X . This relates to the situation where stakes are multiplied by a factor . Inference Rule 2: From X and Y deduce X + Y . This relates to the combination of two gambles. Then, in the usual way, we say that ?  L proves X 2 L, abbreviated to ? ` X , if there is a nite sequence X1; : : :; Xn of gambles in L with Xn = X and each Xi is either an element of ?, an axiom or is produced by an inference rule from earlier elements in the sequence. The logical closure, Th(?), of ?  L is de ned to be the set fX 2 L : ? ` X g. The relation ` is re exive i.e., Th(?)  ?; it is monotonic i.e, ?   ) Th(?)  Th(); it is transitive, i.e, Th(Th(?))  Th(?) (and therefore Th(Th(?)) = Th(?)) and compact i.e, if ? ` X then there exists nite ?0  ? with ?0 ` X . Set of gambles ?  L is said to be nitely-generated if Th(?) = Th(?0) for some nite ?0  ?.

Semantics The set of models M of L is de ned to be the set of probability functions on . For probability function P and gamble X , P(X ) is de ned to be the expected value of X , i.e., P!2 P(!)X (!). We say that probability function P satis es gamble X (written P j= X ) i P(X )  0, that is, if and only if the expected utility is non-negative, i.e., i Walley5 is mainly concerned with the notion of `desirable' gambles which does not quite correspond to the relation ` de ned here. Desirable gambles can be characterized with relation ` , de ned in the same way as `, except the axiom schema is replaced by the following reduced set: X , for any X with minX > 0. With the exception of completeness for nitely generated ?, ` has all the properties of ` given here. i

0

0

we would expect (in the long run) not to lose money from X if P represented objective chances. Note that P satis es the constraint 1P(a1) +


+ k P(ak )   if and only if P j= X where X = 1a1 +


+ k ak ?  is the gamble representing the constraint. We extend j= in the usual way: P satis es set of gambles ? (written P j= ?) if and only if it satis es every gamble X in ?, and the semantic entailment relation j= is de ned by ? j= X i for all models P, [P j= ? ) P j= X ]. The set of probabilities satisfying a given set of gambles ? will be represented as [?]. This set is always convex: it is determined by the linear restrictions associated with the gambles in ?. Theorem 1 (Soundness) For gamble X and set of gambles ?, ? ` X ) ? j= X . We almost have completeness: Theorem 2 (Almost Completeness) For gamble X and set of gambles ?, if ? j= X then ? ` X +  for all  > 0. We also have ? j= X if and only if for all " > 0, ? ` X + "; and also ? j= X + " for all " > 0 if and only if for all " > 0, ? ` X + ". If ? is nitely generated then we have full completeness: ? ` X ) ? j= X .

Lower Previsions and Lower Probabilities Associated with a set ? of gambles is the lower prevision P : L ! IR de ned, for X 2 L by P (X ) = supf : ? ` X ? g. The value P (X ) is the supremum of prices that the agent is willing to pay for X . By Soundness and Almost Completeness, P (X ) = supf : ? j= X ? g, which leads to P (X ) = inffP(X ) : P j= ?g. Lower prevision P is thus the lower envelope of fP : P j= ?g. If a is a proposition, P (a) is usually referred to as the Lower Probability of a. P (a) is the in mum of P(a) over all probability functions P satisfying the constraints represented by ?. The supremum of P(a) over all such probability functions is known as the Upper Probability, P (a), and is given by P (a) = 1 ? P (a). The proof theory can thus generate the optimal upper and lower bounds for the probability of a (or of the expected value of a gamble X ), although it may well be less ecient than other more specialized methods12;13.

Transforming the Frame Assume that f is a mapping relating to sets of possibilities and 0, f : ?! 0. Let L and L be the set of gambles on and 0 respectively. If we have an acceptable set of gambles ? in , then the induced set of gambles in 0, will be 0

?0 = f (?) = fX 0 2 L : X 0  f 2 Th(?)g This rule takes the closure of a set before doing the transformation. The result is always closed. ?0 includes also gambles saying that f ( ) is true. Furthermore it is coherent with the probabilistic meaning, in the sense that it is equivalent to the usual way of transforming probability distributions. We have: 0

[?0] = ff (P) : P 2 [?]g where f (P) is a probability distribution on 0, de ned by f (P)(!0) = P(f ?1 (!0)). If now ?0 is a set of gambles in 0, then the inverse transformation of ?0 will be: ? = f ?1 (?0) = fX 0  f : X 0 2 ?0g

The inverse of a closed set is not always closed. If ?0 is closed then we should take the closure of f ?1(?0 ) to get a closed set. In terms of the associated probabilities this transformation can be expressed as: [?] = fP : f (P) 2 [?0]g

BELIEF REVISION An important problem is how to modify our beliefs (for this logic, a set of gambles) when we receive new information. If the new information is an observation that a proposition a is true then we focus our beliefs to a|see Walley5 for details. In probability this is done by means of conditioning. However, if the new information is an extra set of gambles that the agent nds acceptable (perhaps after further consideration of the problem) then we need some way of incorporating this new set of gambles. This distinction between two types of information is very important in most uncertainty formalisms. The rst type represents information speci c for the situation at hand (the relevant facts)14. The second one represents the background information, the rules valid for generic situations. In the case of classical probability theory both types of information can be handled by means of the conditioning formula. However this can be confusing for other formalisms in which there is no single procedure appropriate for all the situations. This is true in the case of convex sets of probabilities15;11. The incorporation of particular knowledge is usually called focusing and the incorporation of general information, revision. The two give rise to dierent procedures.

Desirable Properties of Expansion and Revision Gardenfors9 considers the operations of expansion and revision of a logically closed set of propositions K by a proposition A (and also contraction, which we do not consider here). Expanding K by A means adding A to the belief set K without retracting any of K ; the result of the expansion is called KA+ . Revising K with new information A means that we add A to the belief set but, if necessary, remove some of K so that the revised belief set KA is consistent. He advances postulates for both these operations. This work trivially generalizes to expanding and revising K by a nite set of propositions ?, by adding the extra condition that K?+ = KA+ and K? = KA where A is the conjunction of the elements in ?. Given this, Gardenfors' postulates can be expressed as follows:

Postulates for Expansion + + (K 1) Th(K? ) = K?+ . (K + 2) ?  K?+. (K + 3) K  K?+ . (K + 4) If ?  K then K?+ = K . (K + 5) If K  H then K?+  H?+ .

(K + 6) For all logically closed K and all ?, K?+ is the smallest set satisfying (K +1){ (K +5).

Postulates for Revision Th(K? ) = K? . ?  K?. K?  K?+ . If K [ ? is consistent then K?+  K? .

(K 1) (K 2) (K 3) (K 4)

(K 5) K? is inconsistent if and only if ? is inconsistent. (K 6) If Th(?) = Th(
) then K? = K
 . (K 7) K?[
 (K?)+
. (K 8) If K? [
is consistent then (K?)+
 K?[
. The postulates in this form can be applied to expanding and revising a logically closed set of gambles K by another set of gambles ? (which we will allow to be in nite). Furthermore, the justi cations of the above postulates9 may also be used for this case. The postulates for expansion determine a unique expansion operator, and, not surprisingly, we get the same result as for the propositional case. Theorem 3 There +is a unique expansion operator satisfying (K +1){(K + 6) (which + sends pairs (K; ?) to K? ) and it is given by K? = Th(K [ ?). The result also holds if we only allow nite ?. As in the propositional case, revision is more complex. It turns out that total pre-orders on K lead to a method of revision which satis es the above postulates. De nition A relation  on a set W is said to be a total pre-order if it is re exive, transitive and for all v; w 2 W , either v  w or w  v (or both). A subset G of W is said to respect  if [w 2 G and w  v ) v 2 G ]. For each logically closed set of gambles, K let K be a total pre-order on K . The meaning od X K Y is that X is less believed or less preferred than Y ii. In this sense, X should be retracted before Y to achieve consistency with the new information ?. For ?  L de ne K? by X 2 K? if and only if either (i) ? is inconsistent, or (ii) there exists G  K such that G respects K , G [ ? is consistent and G [ ? ` X . Theorem 4 The revision operator which maps pairs (K; ?) to K? satis es the belief revision postulates (K  1){(K  8). For classical belief revision there is a representation theorem (Theorem 4.30 of Gardenfors9 ) which shows (roughly speaking) that any revision operator satisfying the postulates can be represented using total pre-orderings on each belief set K . It is not clear that such a representation theorem is possible in this framework. However, if we were to de ne propositions to be sets of probability functions, de ne entailment as , then Theorem 1 of Spohn16 suggests that there will be such a representation theorem between belief revision operators (or non-monotonic inference relations) and total pre-orders on the set of probability functions (with an appropriate well-ordering condition). Unfortunately it seems that this does not lead to a representation theorem for the gambles logic (using the correspondence between logically closed sets of gambles and convex sets of probability functions) because the `spheres' may not be convex16. Example Let = f!1; !2; !3g. A probability distribution on pcan be represented as a point on a equilateral triangle with sides of length equal to 2= 3. The distances of the point to each one of the triangle sides are equal to the probabilities P(!i ), see Fig. 1. Let K = Th(X ), where X is given by X (!1) = 0:5; X (!2 ) = ?0:5; X (!3) = ?0:5. With X we are expressing the inequality P(!1 )  0:5. Now we de ne the following revision procedure: { If ? and K are compatible then, K? = Th(X [ ?). Fig. 2 expresses this revision in terms of the associated convex sets, [K ] and [?]. ii

This is the reverse preorder of the one used by the authors8

P(w ) 3

P(w ) 2

P(w ) 1

Figure 1: One probability [K] [K* ] Γ

[Γ]

Figure 2: Revision in the case of compatibility { If ? and K are incompatible, then consider the family of gambles X given by

X(!1) =  ? 1; X (!2) = ; X (!3) = ; where  2 [0; 1]. Finally determine 0 as the in mum al values  such that X and ? are compatible. The revision will be Th(? [ X0 ). Fig. 3 shows how is the revision in the case of incompatibility: we take only the probabilities from [?] with minimum P(!1). This revision procedure veri es all the revision postulates (K 1){(K 8) however it can not be characterized by a preorder. Let ? = fX0:3g. The revision K? is equal to Th(X0) (the probabilities with P(!1) = 0). It can be proved that there is is no subset G from Th(X ), such that G [ ? ` X0. So it cannot de ned from a preorder.

[K]

[Γ]

[K*Γ ]

Figure 3: Revision in the case of incompatibility

In above example we have shown that there is a revision procedure, which can not be de ned from a preference preorder. Now the question is: Is it such a revision procedure natural? From our point of view it is not. Observe that we are revising a convex set of probabilities [K ] with a new convex set [?]. To solve the incompatibility case, we take the probabilities from [?] which are furthest from the probabilities on K : this makes no sense. There is a natural topology on the space of the probability distributions, and this structure has to be taken into account to de ne reasonable revision procedures. Perhaps there are more reasonable revisions apart from those determined by Theorem 4, but it is clear for us that not all the revisions verifying the eight revision postulates can be accepted. Revisions based on preorders are always compatible with the structure of the set of associated probabilities. The characterization of reasonable revisions, by adding new axioms to (K 1){(K 8), needs further exploration. This class of revisions will be intermediate between the preorder based revisions and those verifying (K 1){(K 8).

Total Preorders and Possibility Distributions de ned on Convex Sets of Probabilities A total preorder, K on a closed set of gambles K can be transformed into a total preorder relation, N , on the complete language L in the following way: { X N Y , if X 2 L ? K . { X N Y when KX j= Y , where KX = fZ 2 K : X K Z g, if X 2 K This total preorder de nes the same revision as the original K and is very similar to a qualitative necessity relation17 on L. There is not a direct translation of qualitative necessity properties because here we do not have a conjunction operator. We could consider that the conjunction of gambles X and Y is the set fX; Y g and to extend N to sets of gambles in the following way: if ?1 , ?2 are sets of gambles then, ?1 N ?2 if and only if 8X 2 L; (X N Z; 8Z 2 ?1) implies (X N Z; 8Z 2 ?2) Making the correspondence that conjunction corresponds to set union, > to ; and ? to L, then we can rewrite the axioms for qualitative necessity as follows A0 A1 A2 A3 C

L N ; and ; 6N L ?1 N ?2 or ?2 N ?1 N is transitive ? N ; If ?1 N ?2 then ?1 [ ?3 N ?2 [ ?3

It is immediate to show that all these axioms are ful lled. It is also veri ed that a set of gambles, ? is equivalent to the set Th(?). Taking into account the semantics of gambles, we can de ne a necessity relationship on the class of closed convex sets of probabilities from M. If H1; H2 are closed convex subsets of M, we say that H1 N H2 if and only if ?1 N ?2, where ?i (i = 1; 2) is the the set of gambles veri ed by every probability in Hi. The axioms for qualitative necessity relationships can now be written as:

A0' A1' A2' A3' C'

; N M and M 6N ; H1 N H2 or H2 N H1 N is transitive ; N H If H1 N H2 then (H1 \ H3) N (H2 \ H3 )

This necessity is only de ned for closed convex sets. It can be extended to a necessity de ned for arbitrary sets as usual. If T1; T2  M, we say that T1 N T2 if and only if 8H1  T1; H1 closed and convex, 9H2  T2; H2 closed and convex, such that H1 N H2 . From this qualitative necessity we can de ne a dual qualitative possibility17, , de ned on the subsets of M. If T1 and T2  M , then we say that T1  T2 if and only if T2 N T1. The qualitative possibility relationship can be expressed as a possibility measure de ned on the set 2M and taking values on the set of equivalence classes of 2M under the equivalence relation T1  T2 if and only if T1  T2 and T2  T1. If C (T ) is the equivalence class of T , and considering the total order on the set of equivalence classes given by: C (T1)  C (T2) if and only if T1  T2, then the mapping  given by (T ) = C (T ) is a possibility measure. To be precise, we have that (T1 [ T2) = maxf(T1); (T2)g In general, not every possibility measure can be de ned from a possibility distribution. For that, we need that former property is veri ed for arbitrary families of sets. That is, [

( Ti) = supf(Ti) : i 2 I g i2I

In our case, this equality is veri ed and then we have that, (T ) = supf(P) : P 2 T g where (P) = (fPg). Next theorem is a characterization of the possibility distributions which can be obtained in this way. Theorem 5  is a possibility distribution associated with a total necessity preorder N in L if and only if there is a mapping , de ned on the set M to a non-trivial completely ordered set D with supremum and in mum, verifying { If P = P1 + (1 ? )P2 ; then (P)  minf(P1 ); (P2)g { If fPngn2IN ! P then (P)  inff(Pn) : n 2 IN g { (P1 )  (P2) if P1 2 [K ]

{ (P1) 6 (P2) if P1 2 [K ] and P2 2 M ? [K ]

and such that being X; Y 2 L, we have

X N Y when ((P )  D; 8P 2 [X ]) implies ((P )  D; 8P 2 [Y ]) Following this theorem, to de ne a revision procedure associated to a belief state K , we only have to determine a possibility distribution on the set of probabilities, verifying that the possibility of the convex combination of two probabilities is always greater that the minimum of the possibilities of the two probabilities we combine. In terms of probabilities the resulting revision can be expressed as follows. For each D in the ordered set D let [K ]D = fP 2 M : (P)  Dg. This de nes a system of spheres similar to Grove's system18. Assume now that the new information is a set of gambles ?. Consider the set of spheres with non-empty intersection with the set [?]: D0 = fD 2 D : [K ]D \ [?] 6= ;g. T In these conditions, [K?] = D2D [K ]D. This intersection is non-empty, when ? is non-empty. In conclusion, to calculate the revision we consider the spheres centered in K with non-empty intersection with ? and take the intersection (in some sense the smallest) of these spheres. 0

Revisions Based on Similarity relationships. Particular Cases In the last section we considered a very general way of de ning revisions. If we want to automatize the process of revision we have to consider more concrete procedures of doing it. A very intuitive way of de ning revisions is by means of similarity relationships. A similarity relationship on M will be a mapping,

S : M  M ?! IR+ verifying the following properties { S (P; P) > S (P; P0) for every P0 6= P. { If P = :P1 + (1 ? ):P2, P0 = :P3 + (1 ? ):P4, (  0), then

S (P; P0)  minfS (P1; P3); S (P2; P4)g { If fPn gn2IN ! P and fP0ngn2IN ! P0 then

S (P; P0)  inffS (Pn ; P0n) : n 2 IN g

Given a closed set of gambles K then the revision based on similarity relationship, S , is the revision determined by the possibility distribution

(P) = S (P; [K ]) = supfS (P; P0) : P0 2 [K ]g It is immediate to show that this possibility distribution veri es the conditions of Theorem 5. A similarity can be de ned from a distance d : M  M ?! IR+0 verifying { d(P; P0 ) = 0 , P0 = P.

{ If P = :P1 + (1 ? ):P2, P0 = :P3 + (1 ? ):P4, (  0), then d(P; P0 )  minfd(P1; P3); d(P2; P4)g { If fPn gn2IN ! P and fP0ngn2IN ! P0 then d(P; P0)  supfd(Pn ; P0n ) : n 2 IN g by means of the expression

1 S (P; P0) = 1 + d(P ; P0 )

The revision in terms of a distance can be expressed in the following way: P 2 [K?] , d(P; [K ]) = inffd(P0; [K ]) : P0 2 [?]g where d(P; [K ]) = inffd(P; P0) : P0 2 [K ]g. Some examples of distances are the following:  d1(P; P0) = P!2 jP(!) ? P0(!)j  d2(P; P0) = P!2 P(!) log( PP((!!)) ) (Kulback and Leibler informational distance) P

0

 d3(P; P0) = !2 (P(!) ? P0(!))2 Example Consider that [K ] has only a single probability distribution, [K ] = fPg and that we want to revise it with a gamble, ? = fa ? 1g. In this case we have a

probability that we want to revise with the additional information: P(a) = 1. Depending on which measure we use we get dierent revisions. By using d1, [K?] is the convex hull of all the probabilities that can be obtained by Lewis imaging19 of P by a. The result in fact is a belief function20;21. In this case, the probability assigned to the complementary of a is not distributed among the elements of a, but assigned to the complete event a. This was proposed by Dubois and Prade11 as an updating rule for probabilities, but with the inconvenience that the result of updating a probability distribution is a family of probabilities not a unique one. By using d2 we obtain precisely the conditional probability to a: [K?] = fP(:ja)g. This justi es probabilistic conditioning as a revision rule22. However, as we are going to see later this is not the unique interpretation of probabilistic conditioning. In fact, this an important source of misunderstandings when working with non precise probabilities. Dierent rules for convex sets of probabilities coincide with probabilistic conditioning when applied to a single probability. All of them are possible extensions of conditioning. The selection of a particular rule depends on the particular situation we are trying to solve. When using d3, [K?] = fP0g, where ( !2a 0 P (!) = 0P(!) + P(a)=jaj ifotherwise where jaj is the number of elements of a. Example Consider now that K is a belief state associated to a belief function, Bel. A belief function21;23 Bel is a mapping Bel : 2 ! [ 0; 1 ] such that there is a mapping m : 2 ! [ 0; 1 ] verifying,

m(;) = 0 P a m(a) = 1 Bel(b) = Pab m(a) m is called the basic mass assignment associated with Bel. If we interpret Bel as lower probabilities, then the associated belief state is K = fa ? Bel(a) : a  g, representing the set of probabilities [K ] = fP : P(a)  Bel(a); 8a  g. Assume that we are revising this knowledge with the gamble, ? = fa?1g, expressing that P(a) = 1. If we use distance d2, then the result of revising is the set of gambles associated with the belief function BelD(:ja), where ) ? Bel(a) BelD(bja) = bel(a1 [?bBel (a) This is the so called Dempster's conditioning of belief functions20. Some authors have disclaimed the use of this rule of conditioning under the upper and lower probabilities interpretation of belief functions. This result shows it can be applied as a revision rule, having a concrete interpretation. In classical probability theory conditioning is at the same time a revision rule (appropriate when we add new general knowledge) and a focusing rule (appropriate when we concentrate on the observations for a particular case). Now, Dempster's conditioning can be considered as a generalization of probabilistic conditioning, but only for revision. Similar results have been reported by Moral and Campos24 and Gilboa and Schmeidler25. The application of Dempster's conditioning to the focusing case has more problems.

FOCUSING A SET OF GAMBLES Let us consider the focusing problem. In convex sets of probabilities, the usual way of focusing is by conditioning all the possible probabilities. If we have a convex set of probabilities H  M, and an observation given by an event a  , the conditioning of H to a is Ha = fP(:ja) : P 2 H g. This conditioning was introduced by Dempster20. The consideration of this formula as a focusing rule can be found in De Campos et al.26 and Dubois and Prade10;11. This is the expression of focusing in terms of convex sets of probabilities. Now we are going to express it in terms of sets of gambles. Assume that K  L is a closed belief state and that we want to calculate the focusing to event a  . Let La be the language associated to event a, that is La is the set of gambles X : a ?! IR de ned on event a. Let us identify the gamble X 2 La with the gamble X 0 2 L given by:

X 0(!) =

(

X (!) if ! 2 a 0 otherwise

Under the consideration that X = X 0, the focusing of K to a can be expressed in a very simple way: Ka = K \ La. The interpretation is very simple. Each gamble expresses some relationship among the probabilities, which is always relative: proportional gambles express the same relationship. To know how to act in the presence of a, we only have to know the relative values of probabilities for the elementary events from a. The following theorem shows that this is correct.

Theorem 6 If Ka = K \ La then [Ka] = [K ]a = fP(:ja) : P 2 K g.

In last theorem is considered that in the focusing process there is a change of language to La, and therefore that the conditional probabilities are de ned on a. If we want to keep the original space L, then the focusing should be expressed as Ka = Th((K \ La) [ fa ? 1g). Another way of focusing is the one proposed by Moral and Campos24. In terms of convex sets of probabilities, it considers that calculating only the convex set Ha loses information. The observation a not only allows us to transform a probability P into P(:ja), but at the same time, assigns a likelihood or possibility value to this probability which is equal to P(a). When the same conditional probability is obtained from dierent probabilities on H , then its possibility value is the maximum of the possibilities corresponding to each one of them. To reproduce this procedure from a syntactic point of view, we consider for each event a, the set a [ feg, where e stands for the contradiction, then after observing a we transform K by means of the mapping,

fa : ?! a [ feg de ned by

(

!2a fa(!) = !e ifotherwise The focusing of K to a will be fa(K ), according to the rules for transforming a frame given in section 2. The correspondence of this procedure with the syntactic one is very simple. Now [fa(K )] is a set of probabilities de ned on a [ feg. We can assign to each one of the probability distributions P 2 [fa(K )] a probability de ned on a, given by: P(!ja) = P(!)=(1 ? P(e)) and a possibility value 1 ? P(e). The set of conditional probabilities and possibility values (with the maximum rule when a conditional probability comes from dierent probabilities) are the same that the ones assigned above. So fa(K ) contains all the relevant information to carry out the focusing rule with possibilities. An interesting result is that this possibility distribution de ned on the set of conditional probabilities veri es the conditions of theorem 5. So it can be a basis for future revisions of the resulting set fa(K ). More concretely we have the following theorem. Theorem 7 If fa(K ) is the result of focusing K by event a, and we de ne on the set of probabilities on a the possibility distribution:

(P) = maxf1 ? P0(e) : P0 2 [fa(K )]; P0(:ja) = Pg then this possibility distribution satis es the conditions of theorem 5 for a set Ka0  La . The set Ka0 is de ned by the conditional probabilities with a maximum value of possibility. If K is the set of gambles associated to a belief function, Bel then the gambles in Ka0 are precisely the ones associated with the Dempster conditioning of the original belief: BelD. We obtain a new interpretation of Dempster's conditioning rule as a focusing rule: it is a focusing rule in which we keep only the probabilities with maximum possibility. That is, at the same time that we make a focusing we do a, perhaps very strong, inference about what are the possible probabilities. This result was also obtained by Gilboa and Schmeidler25.

CONCLUSIONS Revision and focusing procedures for convex sets of probabilities are used for different aims and have dierent but related mechanisms. Revisions for convex sets of probabilities are not completely characterized by the Gardenfors postulates. Some additional work is necessary to characterize correctly coherent revisions for sets of gambles. The use of preorders guarantees that we obtain reasonable revisions, but it may well be the case that other revisions could be considered reasonable too. Focusing and revision rules can be expressed very easily by using the logical language of gambles. These expressions can be used to generate algorithms to calculate these procedures without resorting to the underlying probability distributions which is more complex in most of the cases. In concrete, for the conditioning of all the probabilities in a convex set to an event a, we have shown that in terms of restrictions this can be carried out by calculating the restrictions which have a value of 0 as coecient for the elementary events outside a. A set of restrictions generating this set can be calculated by parametric linear programming.

Acknowledgements The second author was partially supported by a SERC postdoctoral fellowship. This work was also partially supported by ESPRIT II basic research action 6156, DRUMS II. We are also grateful for the use of the computing facilities of the school of CMS, Oxford Brookes University.

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