Possibilistic Causality Consistency Problem with Compound Causal Relationships

The paper addresses uncertain reasoning based on causal knowledge given by two layered networks, where nodes in one layer express possible causes and those in the other are possible effects. Uncertainties of the causalities are given by conditional causal possibilities, which were proposed to express the exact degrees of possibility of causalities. They also have an advantage over the conventional conditional possibilities in the number of possibilistic values that should be given as a priori knowledge. The number of conditional causal possibilities given as knowledge is far smaller than that of conventional conditional possibilities. However, their weakness is that they cannot deal with causalities with compound effects by plural causes, such as synergistic and canceling effects on uncertainty of causalities. The paper discusses this weakness and proposes a solution. Introduction Modeling causal relationships among events, and predicting/diagnosing unknown events from the known are very common tasks in systems analysis. There are many approaches to realize those tasks. However, it is becoming popular to model causalities on causal networks, and to use them for prediction or diagnosis. Belief networks, which are a type of probabilistic network, are attracting a great deal of attention for this purpose [6]. Another type based on conditional causal probabilities is also noteworthy [7]. The conditional causal probabilities were proposed to express precise probabilities of causalities. Probability is a ratio scale of uncertainty. Thus the quantity of its value has an exact meaning of its own, and should be evaluated accurately within its tolerance. In solving real world problems, however, it is frequently the case that not enough data can be gathered to determine the probabilities statistically. In those cases, rough subjective probabilities tend to be used instead. However, it has been indicated that accurate evaluation of prior probabilities is crucial in diagnostic problems, because a small difference in prior probabilities may affect the diagnostic results, namely the probabilistic order of possible causes [8]. There is another scale of uncertainty called possibility [5,12]. Since possibility is essentially an ordinal scale [2], reasoning based on the theory is insensitive to some errors of evaluation of uncertainty. Thus it might be suitable to problems with not enough data to calculate probabilities with high reliability. Possibilistic causal reasoning has recently been studied in this context [3,8-11]. In early days of these studies, conditional possibilities were used to express possibility of causalities [3,8]. These days, however, approaches with conditional causal possibilities, which are a possibilistic version of conditional causal probabilities, have been proposed [9-11]. This is because they can express the precise degrees of possibility of causalities, as well as because the number of conditional causal possibilities given as a priori knowledge is far smaller than that of conventional conditional possibilities [7,9,10]. However, there is a weakness that they cannot deal with causalities with compound effects by plural causes, such as synergistic and canceling effects on uncertainty of causalities [10]. The paper addresses the weakness mentioned above. It discusses how to deal with the compound effects in a causal model, and defines the compound causal model. Then it defines and solves the possibilistic causality consistency problem based on the proposed model. 2 Causal Model and Conditional Causal Possibility 2.1 Asymmetrically-valued causal model Definition 2.1 [7] Let ui (i=1,É,I), v j (j=1,É,J) be events. The presence of v j is dependent on at least one of uis. ui and v j are the cause and effect events, respectively. The single symbol v u j i : denotes a causation event where " ui arises, and ui actually causes v j", and satisfies the next logical formulae: v u v u u v u v v u u v j i j i i j i j j i i j : ( : ) ( : ) ( : ) , ↔ ∧ ↔ ∧ ↔ ∧ ∧ (1) v u u u j i i i : , ∧ ↔ (2) v u v v j i j j : , ∧ ↔ (3) where ∧ and ↔ are conjunction and equivalence, respectively. ui , v j and v u j i : denote negations of ui , v j and v u j i : . Definition 2.2 [11] Let xi, y j be variables taking a value in U u u i i i = { , } , V v v j j j = { , } , respectively. Then xi and y j are the cause and effect. If the following formulae are satisfied, xi and y j follow the asymmetrically-valued causal model (AVC model). Possibilistic Causality Consistency Problem with Compound Causal Relationships, The Fourth Asian Fuzzy Systems Symposium, pp. 89-94 (2000) 90 v v u j i j i ↔ ∨( : ) . (4) v u v u v u False j i j i j i : : : ↔ ↔ ↔ . (5) AVC model is used in most inverse problems of causalities employing causation events [7,9], and eq.(4) is called the mandatory causation assumption. If U u u i i i n = { ,..., } 1 , V v v j j j m = { ,..., } 1 and v v u j k i h j k i h ↔ ∨ , ( : ) as in [10], xi and y j follow the symmetrically-valued causal model [11]. In the case of AVC model, the following is derived from eq. (4). v v u j i j i ↔ ∧( : ) . (6) 2.2 Conditional causal possibility Possibility is one of the scales for measuring uncertainty. Let E1, E2 be two exclusive and exhaustive sets of events, and e1, e2 denote elements of those sets. π( ) e1 and π( ) e2 are marginal possibility distributions on E1 and E2 , respectively. Then the following hold in general [5,12]. π π π ( ) ( ) ( ) e e e e 1 2 1 2 ∧ ≥ ∧ . (7) π π π ( ) ( ) ( | ) e e e e e 1 2 1 2 1 ∧ = ∧ . (8) π π ( ) ( ) e e x x E 1 1 2 = ∧ ∈ ∨ . (9) In the paper, ∧ ∨ ( ) denotes conjunction (disjunction) when used for events, while it denotes min (max) for possibilities. π (e e 1 2 ∧ ) is a possibility distribution of < > e e 1 2 , on E E 1 2 × , and equals to the joint possibility distribution π(e e 1 2 , ) . Using a possibility distribution, a possibility measure is defined as a set function as shown below [12]. Π ({ }) ( ) e e 1 1 = π . (10) Π Π ( ) ({ }) ( ) D e e e D e D = = ∈ ∈ ∨ ∨ 1 1 1 1 π . (11) In the above, D is a subset of E1, and Π ( ) E1 1 = . Suppose that π ( ) e e 1 2 ∨ is defined by π ( ) ({ , | } { , | }) e e e y y E x e x E 1 2 1 2 2 1 ∨ = < > ∈ ∪ < > ∈ Π . (12) Then the next equation is derived. π π π π π ( ) ({ , }) ({ , }) ( ) ( ) ( ) ( ). e e e y x e