An axiomatic framework for belief updates

In the 1940's, a physicist named Cox provided the fir�t formal justification for the axioms of probability based on the subjective or Bayesian interpretation. He showed that if a measure of belief satisfies several fundamental properties, then the measure must be some monotonic transformation of a probability. In this paper, measures of change in belief or belief updates are examined. In the spirit of Cox, properties for a measure of change in belief are enumerated. It is shown that if a measure satisfies these properties, it must satisfy other restrictive conditions. For example, it is shown that belief updates in a probabilistic context must be equal to some monotonic transformation of a likelihood ratio. lt is hoped that this formal explication of the belief update paradigm will facilitate critical discussion and useful extensions of the approach. Introduction As researchers in artificial intelligence have begun to tackle real-world domains such as medical diagnosis, mineral exploration, and financial planning, there has been increasing interest in the development and refinement of methods for reasoning with uncertainty. Much of the work in this area has been focused on methods for the representation and manipulation of measures of absolute belief, quantities which reflect the absolute degree to which propositions are believed. There has also been much interest in methodologies which focus on measures of change in belief or belief updates1, quantities which reflect the degree to which beliefs in propositions change when evidence about them becomes known. Such methodologies include the MYClN certainty factor model [5], the PROSPECTOR scoring scheme [6], and the application of Dempster's Rule to the combination of "weights of evidence" [7]. In this paper. a formal explication of the belief update paradigm is given. The presentation is modeled after the work of a physicist named R.T. Cox. In 1946, Cox [8] enumerated a small set of intuitive properties for a measure of ahsolute belief and proved that any measure that satisfies these properties must be some monotonic transformation of a probability. In the same spirit, a set of properties or axioms that are intended to capture the notion of a belief update are enumerated. It is then shown that these properties place strong restrictions on measures of change in belief. For example, it is shown that the only measures which satisfy the properties in a probabilistic context are monotonic transformations of the likelihood ratio >.(H.E,e) = p(EiH,e)/p(EHl,e), where H is a hypothesis, E is a piece of evidence relevant to the hypothesis, and e is background information. It should be emphasized that the gonl of this axiomization is not to prove that belief updates can only take the form 123 described above. Rather, it is hoped that a formal explication of the update paradigm will stimulate constructive research in this area. For example, the axioms presented here can serve as a tool for the iden(if.ication and communication of dissatisfaction with the update approach. . Given the properties for a belief update, a researcher may be able to pinpoint the source of his dissatisfaction and criticize one or more of the properties directly. In addition, a precise characterization of the update paradigm can be useful in promoting consistent use of the approach. This is important as methodologies which manipulate measures of change in belief have been used inconsistently in the past [9]. Finally, it is hoped that the identification of assumptions underlying the paradigm will allow implementors to better judge the appropriateness of the method for application in a given domain. · Although there has been much discussion concerning the foundations of methodologies which focus on measures of absolute belief [8, 10, 11]. there have been few efforts directed at measures of change in belief. Notable exceptions are the works of Popper [12] and Good [13]. Popper proposed a set of properties or axioms that reflect his notion of belief update which he called corroboration and Good showed that the likelihood ratio >-(H,E,e) satisfies these properties [13]. Unfortunately, Popper's desiderata are somewhat non-intuitive and restricted to a probabilistic context. The axiomization here is offered as an alternative. Scope of the axlomization The process of reasoning under uncertainty can be decomposed into three components: problem formulation, belief assignment, and belief entailment. Problem formulation refers to the process of enumerating the propositions or events of interest as well as the possible outcomes of each proposition. Belief assignmenr refers to the process of constructing and measuring beliefs about propositions of interest. Finally, belief entailment refers to the process of deriving beliefs from beliefs assessed in the second phase. It must be emphasized that most methods for reasoning with uncertainty, including those in which belief updates are central, focus primarily on the third component described above.2 Indeed, it could be argued that a significant portion of the controversy over the adequacy of various methods for reasoning with uncertainty has stemmed from a lack of appreciation of this fact.3 The axiomization of the belief update paradigm presented here similarly restricts its focus to the process of belief entailment. Fundamental properties for a measure of absolute belief Before presenting the axiomization for belief updates, 1t JS useful to consider the properties Cox enumerated for a measure of absolute belief. This discussion will help motivate the characterization of measures of change in belief as it is similar in spirit. In addition, several of the properties for a measure of absolute belief will be needed for the explication of the belief update paradigm. The first property proposed by Cox concerns the nature of propositions to which beliefs can be assigned. He asserted that propositions must be defined precisely enough so that it would be possible to determine whether a proposition is indeed true or false. That is, a proposition should be defined clearly enough that an all-knowing cfainoyant could determine its truth or falsehood. This requirement will be called the clarity property.4 A .second property asserted by Cox is that it is possible to assign a degree of belief to any proposition which is precisely defined. This property will be termed the completeness property. Cox also asserted that a measure of belief can vary continuously between values of absolute truth and falsehood and that the continuum of belief can be represented by a single real number. For definiteness, it will be assumed that larger numbers correspond to larger degrees of belief. The use of a single real number to represent continuous measures of belief will be called the scalar continuity property. Another fundamental assumptiOil made by Cox is that the degree of belief for a proposition will depend on the current state of information of the individual assessing the belief. To emphasize this, the term Pie, read "P given e," will be used to denote the degree of belief in proposition P for some individual with information e. This assumption will be termed the context dependency property. Now consider two propositions P and Q. If P and Q are logically equivalent then Pie = Qle. That is, if P Is true only when Q is true and vice-versa, an individual should believe each of them with equal conviction. Thus, for example, it must be that XYie = YXIe where XY denotes the proposition "X AND Y." This axiom will be called the consistency property. Another property asserted by Cox is that the belief in the conjunction PQ should be related to the belief in P alone and to the belief in Q given that P is true. Formally, Cox proposed that there exists some function F such that POle = F(Pie. QIPe). ( 1) The function is asserted to be continuous in both arguments and monotonically increasing5 in each argument when the other is held constant. This property captures the notion that individuals commonly assign belief to events conditioned on the truth of another. This property will be termed the hypothetical conditioning property. Finally, Cox asserted that the belief in �P (not P) should be related to the belief in P. Formally, he asserted that there should be some function G such that �PIE = G(Pie). (2) The only restrictions placed on G are that it be continuous and monotonically decreasing. This assumption will be called the complementarity property. After enumerating these properties, Cox proved that any measure which satisfies them must also satisfy the relations: 0 :5 H(Pie) :5 (3) H(TRUEie) (4) H(PQie) = H(Pie) · H(QIPe) (product rule) ( 5) H(Pie) + H(�Pie) = 1. ( s um rule) (6) where H is a monotonically increasing function. However (3) (6) implies that H(Pie) is a probability. That is. (3) 124 ( 6) correspond to the axioms of probability theory. Therefore, Cox demonstrated that if one accepts the above properties, one must accept that probability is the only admissible measure of absolute belief. Cox's proof is simple and elegant. The reader is urged to consult the original work to gain a better appreciation of the argument. The work also contains an interesting discussion by Cox arguing for each of the properties he describes. In the sections to follow, an argument analogous to Cox's for belief updates is presented. As mentioned above, there will be little effort made to justify the properties enumerated. Instead, it is hoped that this exposition will foster constructive discussion about the usefulness of the update paradigm. Fundamental properties for a measure of change in belief . Suppose an individual with background information e has a belief in some hypothesis H for which a piece of evidence E becomes known. The basic assumption of the update par�digm is that a belief update, denoted U(H,E,e), in conjunction with the prior belief, Hie, is sufficient for determining the posterior belief HIEe. More formally, it is assumed that there exists some function f such that HIEe = f(U(H,E,e), Hie). (7) In the paradigm, the quantities U(H.E.e), Hie. and HIEe are all single real numbers. In addition, it is required that the function f be continuous in both arguments and that f be monotonically increasing in each argument when the other is held constant. Equation (7) is the definition of a belief update. Note that only the context dependency property and the scalar continuity property for a measure of absolute belief have been assumed in this definition. It is useful to view the function f in (7) as an updating procedure which operates on a prior belief and returns a poste�ior belief. The procedure, in turn, is parameterized by the smgle parameter U(H,E,e), a function of the hypothesis being updated, the evidence producing the update, and �he ba�kg_round. info.rmation in which the update takes place. Th1s IS depleted 111 the upper diagram of Figure 1. For comparison, the Bayesian conditioning scheme is represented schematically in the lower diagram of the same figure. Corresponding to the updating procedure in the belief update paradigm is the axiomatic engine of probability theory. The axiomatic engine, in turn, is "parameterized" by the full joint distribution. Inputs to the Bayesian updating procedure include the propositions of interest and outputs consist of beliefs relating to these propositions. An important difference between the two approaches is illustrated in the figure. In the Bayesian theory, the process of updating is implicit; it is a matter of course that the belief in a given proposition changes when the conditioning propositions are modified (recall Cox's context dependency property). In contrast, the process of updating is made explic_it in the update paradigm. As a consequence, the Bayes1an scheme can treat hypothesis and evidence symmetrically while the update approach cannot. For example, the calculation of p(EIHe) in the Bayesian approach is no different in principle then the calculation of p(HIEe). In the update approach, however, the roles of evidence and hypothesis would have to be exchanged in order to implement the calculation of p(EiHe). In addition to the definition above, there are two funda�ental proper�ies that are ascribed to belief updates. The f1rst property IS analogous to the consistency property for absolute beliefs. It is assumed that if the arguments of a belief update are logically equivalent, then the belief updates must have the same value. That is, if H1 ""' H2, E1 = E2, I I I