A Logical View of Probability

Transcription

1 A Logical View of Probability Nic Wilson 1 and Serafn Moral 2 Abstract. Imprecise Probability (or Upper and Lower Probability) is represented as a very simple but powerful logic. Despite having a very dierent language from classical logics, it enjoys many of the most important properties, which means that some extensions to classical logic can be applied in a fairly straightforward way. The logic is extended to allow qualitative grades of belief, which can be used to represent degrees of caution, and this is applied to create theories of belief revision and non-monotonic inference for probability statements. We also construct a theory of default probability which is based on a variant of Reiter's default logic this can be used to express and reason with default probability statements. 1 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 justications 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 justication, 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 suchas0:6 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 inmum values of the probabilities of propositions (known as Upper and Lower Probabilities) given the constraints. Early work on this includes Boole [1] and Good [2] and early appearances in the Articial Intelligence literature include [3,4]. Walley's Imprecise Probability [5] 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 Smith [6] and Williams [7]. 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 section 2, we dene this logic with a language, very simple proof theory and semantics. The results of this section all follow fairly easily from results in [5]. 1 Department of Computer Science, Queen Mary and Westeld College, Mile End Rd., London E1 4NS, UK 2 Departamento de Ciencias de la Computacion e I. A., Universidad de Granada, Granada - Spain Gambles can represent a surprisingly varied set of probability statements. Another way of viewing gambles is as linear conc 1994 N. Wilson and S. Moral ECAI th European Conference on Articial Intelligence Edited by A. Cohn Published in 1994 by John Wiley & Sons, Ltd. Apart from being a simple and elegant way to express this theory of probability, there are other benets of expressing it as a logic. It brings into the logician's domain this semantically very well founded, fairly well-behaved and expressive representation of beliefs. Because the logic has many of the properties of classical logics, it means that augmentations of classical logic can be applied relatively easily to this logic. Imprecise Probability does not distinguish caution from ignorance in section 3 we look at a way of extending the theory to allow qualitative grades, which can be used to represent degrees of caution. Like classical logic, it is very conservative, and is monotonic. It therefore seems natural to look at extensions which tentatively allow stronger conclusions to be drawn, but avoid inconsistency. Three examples of this are given in section 4, work on belief revision and non-monotonic inference relations is extended to this logic, which leads to ways of resolving inconsistencies, and in section 5, a version of Reiter's Default Logic is applied, which allows more complex tentative assumptions. 2 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 dened 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 dened pointwise, so that for gambles X and Y, for each! 2, (X ; Y )(!) =X(!) ; Y (!). For a we dene gamble a to be the indicator function of a, so that a(!) =1if! 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)(!) =0if! 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

2 straints 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(a 1)++ k Pr(a k ) is represented by the gamble X = 1a k a k ;. The constraint 1 Pr(a 1)++ k Pr(a k ) is equivalent to the constraint (; 1)Pr(a 1)++(; k )Pr(a k ) ; and so is represented by the gamble ;X. Hence 1 Pr(a 1)++ k Pr(a k )= is represented by the pair of gambles fx ;Xg. 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 dene 3 an inference relation `, where for ; L and X 2L,;`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 dene ` will be justied by the semantics (with probability functions as models) given later. However, as Walley shows, they can be justied 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 ; Lproves X 2L, abbreviated to ; ` X, if there is a nite sequence X 1... X n of gambles in L with X n = X and each X i 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 ; Lis dened to be the set fx 2L :;`Xg. The relation ` is reexive 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 ; Lis said to be nitely-generated if Th(;) = Th(; 0 ) for some nite ; 0 ;. 3 Walley [5] is mainly concerned with the notion of `desirable' gambles which does not quite correspond to the relation ` dened here. Desirable gambles can be characterised with relation `0, 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 ;, `0 has all the properties of ` given here. Semantics The set of models M of L is dened to be the set of probability functions on. For probability function P and gamble P X, P(X) is dened to be the expected value of X, i.e., P(!)X(!). We say that probability function P satis-!2 es gamble X (written P j= X) ip(x) 0, that is, if and only if the expected utility is non-negative, i.e., i we would expect (in the long run) not to lose money from X if P represented objective chances. Note that P satises the constraint 1P(a 1)+ + k P(a k ) if and only if P j= X where X = 1a k a k ; is the gamble representing the constraint. We extend j= in the usual way: P satises set of gambles ; (written P j= ;) if and only if it satises every gamble X in ;, and the semantic entailment relation j= is dened by ; j= X i for all models P, [P j= ;) P j= X]. 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 ">0if and only if for all ">0, ; ` X+". If ; is nitely generated then we have full completeness: ; ` X ) ; j= X. Corollary For ; L, ; `? () ; j=?. If ; j=? then we say that ; is inconsistent otherwise ; is said to be consistent. If ; is inconsistent then ; ` X for any gamble X. The relation j= is reexive, transitive, monotonic, but not compact (consider ; = fn(a ; 1+ 1 n ):n2ing for some nonempty proper subset a of ). However, the above corollary means that it does satisfy the related property `If ; j=? then there exists nite ; 0 ; with ; 0 j=?'. Lower previsions and lower probabilities Associated with a set ; of gambles is the lower prevision P : L!IR dened, for X 2Lby 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 :Pj= ;g. Ifa is a proposition, P (a) is usually referred to as the Lower Probability of a. P (a) is the inmum 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 specialised methods such as [8,9]. 3 GRADED SETS OF GAMBLES The gambles that an agent nds acceptable can depend strongly on how cautious they are. For example, suppose I (the rst author) know that Joe lives in Cowley Road, Oxford (which has Knowledge Representation 387 N. Wilson and S. Moral

3 100 houses) but I've no idea which one. Let = f g refer to the set of possible houses that Joe could live in. Now suppose Bob asks me how much I'd be prepared to pay for the gamble corresponding to the proposition a = nf51g (and let's say 1 utile corresponds to 1000 pounds). I might pick P (a) =0:99, so that I would accept the gamble a ; for any <0:99 or I might consider that I don't know enough about a to bet against Bob and refuse to bet, or only accept extremely favourable odds, leading to P (a) about 0. What I decide here could be extremely arbitrary. The theory does not distinguish between caution and ignorance. A lower prevision/probability P may be close to being Bayesian because the knowledge of the agent is fairly complete, or because the agent is bold in her assessments (perhaps she is used to giving Bayesian beliefs). To deal with this problem we could use graded sets of gambles. A graded set of gambles 4 is a function from a set WLto a set D, where D has a total order on it. (A total order on D is a reexive, transitive relation, such that for all 2 D, either or, and only both if =.) For graded set of gambles g, and X Y 2W, g(x) <g(y ) means that X is more rmly believed than Y. For graded set of gambles g and 2 D, let g = fy 2W : g(y ) g. We write g j= (X ) ifg j= X, and g ` (X ) ifg ` X. The statement g j= (X ) can be thought of as saying that the agent, when being cautious to degree, nds X acceptable. For example D might be fcautious boldg with cautious < bold. fx : g(x) = cautiousg would be the more cautious and more deeply held beliefs, fx : g(x) = boldg would be more tentative beliefs. In the next section we will see how graded sets of gambles give rise to very well-behaved systems of belief revision and non-monotonic inference. 4 BELIEF REVISION AND NON-MONOTONIC INFERENCE RELATIONS An important problem is how to revise 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 condition our beliefs by a see [5] for details. 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. Desirable properties of expansion and revision Gardenfors [11] 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 K + A. Revising K with new information A means that we add 4 There is a strong relationship between graded sets of gambles and fuzzy probabilities and Gardenfors and Sahlin's unreliable probabilities [10] (using the correspondence between logically closed sets of gambles and convex sets of probability functions). 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 generalises to expanding and revising K by a nite set of propositions ;, by adding the extra condition that K + = ; K+ A 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 (K 1) Th(K;)=K ;. (K 2) ; K;. (K 3) K; K +. ; (K 4) If K [ ; is consistent then K + ; K;. (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 innite). Furthermore, the justications of the above postulates [11] 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 satises the above postulates. A relation on a set W is said to be a total pre-order if it is reexive, transitive and for all v w 2W, either v w or w v (or both). A subset G of W is said to respect if [w 2Gand v w ) v 2G]. For each logically closed set of gambles, K let K be a total pre-order on K. For ; Ldene K; by X 2 K; if and only if either (i) ; is inconsistent, or (ii) there exists GK such that G respects K, G[; is consistent and G[; ` X. Theorem 4 The revision operator which maps pairs (K ;) to K; satises the belief revision postulates (K 1){(K 8). For classical belief revision there is a representation theorem (Theorem 4.30 of [11]) 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 dene propositions to be sets of probability functions, dene entailment as, then Theorem 1 of Spohn [13] suggests that there will be such a representation theorem between belief revision operators (or Knowledge Representation 388 N. Wilson and S. Moral

4 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' [11] may not be convex. Non-monotonic inference relations Makinson and Gardenfors [12] have shown that the belief revision postulates can be translated into properties of nonmonotonic inference relations between propositions. Such a non-monotonic inference relation j can be trivially extended to allow nite sets of propositions on the left hand side, by dening, for nite ;, ; j X () A j X where A is the conjunction of the propositions in ;. Let C(;) = fx :;j Xg. Equivalent versions of these properties of non-monotonic inference relations can be reached from the belief revision postulates (K 1){(K 8) by replacing K; by C(;), so that K = K gets replaced by C( ) (and using theorem 3). For example, ( j 8) is the property If C(;) [ is consistent then Th(C(;) [ ) C(; [ ), which is known as Rational Monotony. Again, in this form these properties can be interpreted as desirable properties for a non-monotonic inference relation on gambles. Theorem 5 Let be a total pre-order on L. Dene inference relation j as follows. For (possibly innite) ; Land X 2L, ; j X holds if and only if (i) ; is inconsistent or (ii) there exists GLsuch that G respects, G[; is consistent and G[; ` X. Then j satises non-monotonic inference relation postulates ( j 1){( j 8). Total pre-orders on sets of gambles are very closely related to graded sets of gambles. From graded set of gambles g on WLwecan dene a total pre-order on W by X Y () g(x) g(y ). (Conversely, if is a total pre-order on W, then dening equivalence relation by X Y () X Y and Y X, and letting D = W=, the total preorder induces a total order on D, and we can dene graded set of gambles g by g(x) = () X 2.) Therefore graded sets of gambles lead to well-behaved belief revision operators and non-monotonic inference relations. 5 DEFAULT PROBABILITY Imprecise probability is very conservative so that the conclusions can be disappointingly weak (especially after conditioning), giving us little information about which propositions are likely or unlikely. For example, constraints P(bja) = 1 and P(cjb) 0:999 lead to P (cja) = 0 since there is a probability function P which satises the two constraints and P(cja) = 0. If, to counter this, the agent makes very bold probability statements then there is a danger of inconsistency, which is even less useful. It is therefore desirable to have a theory of default probability where some of the probability statements are labelled as defaults, and could be retracted if they lead to inconsistency. It is also important to be able to make default independence assumptions, for example, if an expert does not know of any dependencies between two variables it is sometimes natural to assume, by default, that they are independent. A default logic We use an amended version of Reiter's default logic [14] because it has some nicer properties. 5 A default rule is a string of the form A : B=C with A B C nite subsets of L. This rule is intended to mean something like `if all of the elements of A are known and B is consistent with what we know then deduce C'. A default theory is a pair (D W) where WLis known as the facts and D is a set of default rules, which we will assume to be labelled, for some set, A i : B i : i 2 : C i For any dene Th (W) to be the (unique) smallest set ; Lsuch that ; W, Th(;) = ;, and if i 2 and ; A i then ; C i. For, if we allow ourselves to use inference rules A i=c i for i 2, then we can deduce any formula in Th (W) (and no others). However, if S Bi is now inconsistent then this i2 would seem to contradict the intention of the default rules. This motivates the denition of -consistent. Let = (D W) be a default theory. is said to be -consistent if Th (W) [ S i2 Bi is consistent. Dene C = f : is -consistentg. E is said to be an extension of if E =Th (W) for some maximal in C. In default logic the extensions are intended to be the different possible completions, using the default rules, of an incomplete set of facts about the world. In practice, extensions would not usually be calculated, but instead we might generate pairs (X ) where X 2 Th (W ), and we are interested in only those where is -consistent a simple monotonic logic governs the generation of such pairs, see [15]. Extensions could also be dened using j= instead of `. The corollary to Almost Completeness implies that C will be the same, so the subsequent extensions will be almost the same. A type of ordering on the defaults can easily be represented with this approach, which both increases the expressiveness of the default logic, and cuts down the number of extensions. Let be a pre-order (i.e., a reexive and transitive relation) on. Let C = f : is -consistent and respects g, where respects i for all i j 2, such that i j, [j 2 implies i 2 ]. Extensions can be dened as before. 6 If is a total pre-order, then there is always a unique extension. This system can then be seen to generalise the non-monotonic inference relations generated in section 4. Applications of default probability W will contain the gambles of which the agent is condent. Less reliable gambles are represented using default rules with A i = B i =. For example, suppose one wanted to combine the probabilistic judgements of a number (n) of experts, where we elicit from expert i (for each i =1... n) a set of gambles C i that they nd acceptable. This would give rise to a set D 5 In particular, it has a default proof theory, the sceptical inference relation is cumulative, it is semi-monotonic and any default theory has an extension for details and motivation see [15,16]. 6 Semi-monotonicity now does not usually hold, but Cumulativity does and it is still the case that every default theory has an extension. Knowledge Representation 389 N. Wilson and S. Moral

5 of default rules f : =fc ig : i =1... ng. In this way, the judgements of an expert are viewed as being of the same degree of reliability, and the judgements of dierent experts as being of incomparable degrees of reliability. If we had extra information about the relative reliabilities of the dierent experts (or of the dierent gambles that a single expert nds acceptable) then that could be incorporated in. Default inference rules can be represented by default rules with B i =. These are useful for default structural judgements, such as independence assumptions. A single independence assumption requires a schema of inference rules it seems natural to put each of these rules at an equivalent position in the pre-order (i j for each pair of rules i j in the schema) so that if one of these inference rules is rebutted, they all are. Rules with B i 6= might be used to express default statements such as `assume Pr(a) 0:6 unless we nd that Pr(c) < 0:1', using the default rule : fc ; 0:1g=fa ; 0:6g. 6 CONCLUSIONS AND DISCUSSION We have dened a very simple, but powerful, logic of probability. It would be interesting to explore connections with other, more complex, logics of probability such as those in [17,18,19,20], and to look at ways in which this logic could be made more expressive. One way is to allow to be innite, and many of the properties do still follow from results in [5]. It has been shown that theories of belief revision and nonmonotonic reasoning can be extended easily to this logic of probability statements. We have not, however, used any of the special structure of this logic. For example, there are natural metrics that can be put on the set of probability functions, and these might be used to help generate grades in a graded set of gambles. Alternatively, grades for gambles might be calculated from a conditional convex set of probabilities, as in [21], where each probability function is assigned a degree of possibility. Most importantly, for any of these systems to be of practical use, tractable subsystems must be identied for example, we need to look for natural and tractable types of default assumptions, and types of graded gambles that lead to ecient methods of probabilistic belief revision. ACKNOWLEDGEMENTS The rst author is supported by a SERC postdoctoral fellowship. This work was also partially supported by ESPRIT II basic research action 3085, DRUMS II. We are also grateful for the use of the computing facilities of the school of CMS, Oxford Brookes University. REFERENCES [1] G. Boole, An Investigation of the Laws of Thought on which are founded the Mathematical Theories of Logic and Probabilities, Macmillan, London, Reprinted in 1958 by Dover. [2] I. J. 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