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Possibilistic causality consistency problem based on asymmetrically-valued causal model
Fuzzy Sets and Systems 132 (2002) 33 – 48
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Possibilistic causality consistency problem based on asymmetrically-valued causal model Koichi Yamada ∗ Department of Management and Information Systems Science, Nagaoka University of Technology, 1603-1 Kami-tomioka, Nagaoka, Niigata 920-2188, Japan Received 11 January 2000; received in revised form 20 February 2002; accepted 22 February 2002
Abstract 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 e/ects. Uncertainties of the causalities are given by conditional causal possibilities, which were proposed to express the exact degrees of possibility of causalities. The expression of the uncertainty also has 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. The paper starts with the de1nition of a causal model called asymmetrically-valued causal model. The conditional causal possibilities are de1ned on the causal model, and their mathematical properties are discussed. Then, the paper de1nes the possibilistic causality consistency problem based on the proposed model and shows how to solve the problem. The discussed problem is one to calculate the conditional possibility of a hypothesis about presence and absence of c 2002 Elsevier Science B.V. All rights reserved. unknown events when the states of some other events are known. Keywords: Possibility theory; Causal reasoning; Causal network; Uncertain reasoning
1. Introduction Modeling causal relationships among events, and predicting or diagnosing unknown events from the known are very common tasks in systems analysis. The language ordinarily used to model the causalities in the 1eld of engineering is numerical formulae. However, it is also popular to model causalities on causal networks, when the objective system is complex and di:cult to model with numerical formulae. Probabilistic causal networks could be e/ective, especially when the causalities are uncertain. Belief networks (BNs), which are a type of well-established probabilistic networks, are attracting a great deal of attention for this reason [15,16]. However, it is reported that their application is limited [14] in spite of the popularity of the theoretical research due to the fact that problems of learning BNs and probabilistic inference using them are NP-hard [2,3,14], while the recent research is focusing on reduction of the complexity [12,14]. BNs also have a problem to need a huge number of conditional probabilities as a priori knowledge; as many as combinations of states of all possible causes for each e/ect. ∗
Tel.=fax: +81-258-47-9359. E-mail address: [email protected] (K. Yamada).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 0 9 9 - 4
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
A possible alternative of the ordinary BN is a probabilistic causal model based on conditional causal probabilities proposed by [17,18], if the objective system can be modeled by the “noisy-or-gate” where causes generate e/ects disjunctively [16]. In fact, it seems that many diagnostic problems in industry can be modeled by the noisy-or-gate, considering that a more restrictive assumption such as single fault assumption has been common in the 1eld. One of the advantages of the model is that it needs only as many conditional causal probabilities as the number of possible causes for each e/ect. The conditional causal probabilities are those devised to express the exact probabilities of causalities and the same as causal powers studied in psychological literature [1,11]. One of the reasons why they are studied in the 1eld of psychology might be that they reJect the probabilities of causalities that a human recognizes in her mind more accurately than conditional probabilities [1,18]. Thus the use of conditional causal probabilities is more appropriate than the conditional probabilities when we cannot help giving the subjective probabilities instead of statistical ones [18]. Now, probability is a ratio scale of uncertainty. 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 di/erence in prior probabilities may a/ect the diagnostic results, namely the probabilistic orders of possible causes [23]. There is another scale of uncertainty called possibility [6,13,24]. Since possibility is essentially an ordinal scale [7,8], reasoning based on the theory is insensitive to some errors of evaluation of uncertainty. Thus it might be suitable for problems with not enough data to calculate reliable probabilities. Possibilistic causal reasoning has recently been studied in this context [9,23,19,20,21]. In early days of these studies, conditional possibilities were used to express uncertainty of causalities, as conditional probabilities are [23]. Nowadays, however, approaches with conditional causal possibilities, which are a possibilistic version of conditional causal probabilities, have been proposed [19,20,21]. 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 [19,20]. Dubois and Prade also proposed a model of causal reasoning with substantially the same representation of uncertainty of causalities in [9]. The model is expressive for uncertainty in the sense that it handles uncertainty of observation as well as of causalities with two types of representation: “more or less certainly true” and “more or less certainly false” (equivalently, “more or less certainly true” and “more or less possibly true”). Then, the model was expanded in [4], by combining binary states of e/ects with fuzzy sets on numerical attributes. In those models, however, the available observation is limited to e/ects (manifestations) and the hypothesis is to causes (disorders). The paper addresses uncertain reasoning on a two-layered causal network with the conditional causal possibilities. More speci1cally, it de1nes the asymmetrically-valued causal model, a type of causal model with causation events used in [19,21]. Then, it de1nes and solves the possibilistic causality consistency problem based on the proposed model. It is a problem of calculating the conditional possibility of a hypothesis about presence=absence of unknown events, when the states of some other events are known [21]. This problem includes, as special cases, an inverse problem of causalities which calculates the conditional possibility that a set of causes are present and the others are absent given some observed e/ects [23,19]. It also includes a causality analysis problem calculating the conditional possibility of presence=absence of each unknown event when the states of some other events are known [20]. BrieJy, the paper is an expanded one of [19] and [21] in the sense that (1) the causality consistency problem includes the inverse problem of causalities, (2) the possibility of negation of the hypothesis is calculated as well as the possibility of the hypothesis, and (3) some proofs in the papers are re1ned mathematically to be correct.
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
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2. Asymmetrically-valued causal model The section de1nes cause, e/ect, and causation events proposed by [17,18] and the asymmetrically-valued causal model introduced in [21]. Denition 2.1. Let U = {ui | i = 1; : : : ; I } and V = {vj | j = 1; : : : ; J } be two sets of primitive events. U and V are disjoint. The presence of vj is dependent on at least one of ui . Thus, ui and vj are the cause and e4ect (events), respectively. The presence and absence of events are described by classical propositions; a proposition representing presence of an event is denoted by the event itself (e.g. ui , vj ), and one representing absence is by its negation (e.g. ui , vj ). The single symbol vj : ui denotes a causation event (or its presence) that means “ui arises, and ui actually causes vj ”. The causation events satisfy the next logical formula: vj : ui → ui ∧ vj ;
(1)
where ∧ and → are conjunction and implication, respectively. The set of all possible causation events is denoted by V : U = {vj : ui | vj ∈ V; ui ∈ U }. Then, following formulae are easily derived from formula (1). vj : ui ↔ (vj : ui ) ∧ ui ↔ (vj : ui ) ∧ vj ↔ (vj : ui ) ∧ ui ∧ vj ;
(2)
vj : u i ∧ ui ↔ ui ;
(3)
vj : u i ∧ vj ↔ vj ;
(4)
where ↔ denotes equivalence. Peng and Reggia state that the occurrence of causation event vj : ui represents the realization or instantiation of the causal link between ui and vj [18]. They also explain that it is a distinct type of basic event that cannot be expressed by a logical expression of events ui and vj . It is understandable considering the situation, where vj : ui is false even if ui ∧ vj is true. The situation arises if ui and vj are present, but vj is caused by some other cause. Then the asymmetrically-valued causal model is de1ned using the causation events. Denition 2.2. Let Ui = {ui ; ui } and Vj = {vj ; vj }. If the following formula is satis1ed, ui (or Ui ) and vj (or Vj ) follow the asymmetrically-valued causal model (AVC model). vj ↔ (vj : ui ); (5) i
where ∨ denotes disjunction. The above de1nition, together with formula (1), derives the next formula. vj → ui :
(6)
i
Thus, if some e/ect is present, some cause must be present. The AVC model is used in most inverse problems of causalities employing causation events [17,18,19], and formula (5) is called the mandatory causation assumption. If Ui = {ui1 ; : : : ; uin }, Vj = {vj1 ; : : : ; vjm } and vjk ↔ i; h (vjk : uih ) as in [20], uih (or Ui ) and vjk (or Vj ) follow the symmetrically-valued causal (SVC) model [21]. In the case of the SVC model, any element in Vj cannot be present unless it is caused by a cause or
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
some causes. For example, suppose that Vj is a set of possible skin colors of chameleon j and that Ui is a set of possible colors of object i around the chameleons. The positions of the chameleons and the objects are supposed to be 1xed. If the color of a chameleon j is always caused by a color or some colors of the objects in a disjunctive way, Ui and Vj follows the SVC model. Note that there can exist a binary SVC model: the model with Ui = {ui1 ; ui2 }, Vj = {vj1 ; vj2 } and vjk ↔ i; h (vjk : uih ), (h; k = 1; 2). In this case, ifui1 and ui2 (vj1 and vj2 ) are regarded as ui and ui (vj and vj ) respectively, vj ↔ i {(vj : ui ) ∨ (vj : ui )} and vj ↔ i {(vj : ui ) ∨ (vj : ui )} hold. This means that absence of ui may cause vj and that absence of vj may be caused by presence or absence of ui . In the AVC model, however, vj is never caused by any ui as known from the mandatory causation assumption. In addition, vj is not caused by any cause or its negation di/erently from the SVC, and the following formula holds. vj ↔ (vj : ui ): (7) i
In the rest of this paper, we assume Ui = {ui ; ui } and Vj = {vj ; vj }, and that they follow the AVC model. 3. Conditional causal possibility Possibility is one of the scales for measuring uncertainty [6,24]. Let (s i ), i = 1; : : : ; n be a possibility distribution on a universal set S = {s1 ; : : : ; sn }. The distribution must satisfy i (si ) = 1, where ∨(∧) denotes max (min) when used for possibilities, while it denotes disjunction (conjunction) for propositions in the paper. Then, the possibility measure (A), A ⊆ S could be de1ned using the distribution. (si ); (8) (A) = si ∈A
where (A) = 0 if A is the empty set. In Eq. (8), the argument of (•) is a subset of the universe S. However, in the paper, notation such as ( si ∈A si ) is used instead in order to handle possibility of events denoted by logical expressions. Thus, possibility of an event corresponding to a subset with a single element si is denoted by (si ) instead of ({si }). Namely, (si ) = (si ) holds in the notation. In the rest of this paper, possibility of si is denoted by (si ) for convenience. Let ek (k = 1; 2; 3; : : :) be an event representing ui or vj : u. Both of them are elementary events in the sense that they cannot be de1ned by any logical expression of the other events. vj is a composite one, because it can be de1ned by formula (5). Then, let ek∗ be ek or ek , and (ek∗ ) be a possibility that is equal to a marginal possibility distribution (ek∗ ) on Ek = {ek ; ek }. Note that the distributions are de1ned on {ui ; ui } or {vj : ui ; vj : ui } in the paper, but not on U nor V : U . Due to formulae (2) – (4), (vj : ui )6(vj ), (vj : ui )¿(ui ) and (vj : ui )¿(vj ) must be satis1ed. Let (e1∗ ∧ e2∗ ) be a possibility of the joint event e1∗ ; e2∗ , and be equal to the joint possibility distribution (e1∗ ; e2∗ ) on E1 × E. Then, possibility (e1∗ ) is equal to the maximum of (e1∗ ∧ e2 ) and (e1∗ ∧ e2 ). (e1∗ ∧ x): (9) (e1∗ ) = x∈{e2 ;e2 }
For conditional possibility, it is assumed that (d | e) ∨ (dO | e) = 1 holds for any events e and d. The relation between joint and conditional possibilities is given by (e1∗ ∧ e2∗ ) = (e1∗ | e2∗ ) ∧ (e2∗ );
(10)
which was 1rst introduced in [13] as the equation corresponding to the multiplicative law of probability.
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
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Suppose that (e1∗ ∧ e2∗ ) and (e2∗ ) are known and (e1∗ | e2∗ ) is unknown in Eq. (10). Though the solution is not unique in this case, the paper chooses the largest one as its solution according to the idea of the principle of the maximal speci1city or the least speci1c solution [5,8,10]. In the case of Eq. (10), the principle gives the next solution. (e1∗ ∧ e2∗ ); if (e2∗ ) ¿ (e1∗ ∧ e2∗ ); ∗ ∗ (e1 | e2 ) = (11) 1; if (e2∗ ) = (e1∗ ∧ e2∗ ): Then, the next equality is easily obtained using Eq. (11). (e1∗ ∧ e2∗ | e3∗ ) = (e1∗ | e2∗ ∧ e3∗ ) ∧ (e2∗ | e3∗ ):
(12)
Now, if Eq. (13) holds, it is said that (e2∗ ) is possibilistically independent of e1∗ [13]. When Eq. (14) is satis1ed, e1∗ and e2∗ are noninteractive [24]. (e2∗ | e1∗ ) = (e2∗ );
(13)
(e1∗ ∧ e2∗ ) = (e1∗ ): ∧ (e2∗ ):
(14)
e1∗ and e2∗ are noninteractive, if e2∗ (e1∗ ) is possibilistically independent of e1∗ (e2∗ ). For the correspondence between the logical and the possibilistic expression, it is de1ned that any event e is true (present) i/ (e) = 1 and (e) O = 0. I/ (e) = 0 and (e) O = 1, e is false (absent). Then, it is easily obtained that (e | e) = (eO | e) O = 1 by substituting e for e1∗ and e2∗ in Eq. (11). Furthermore, (e | e) O =0 holds except the case where (e) O = 0, and (eO | e) = 0 when (e) = 0. Now, conditional causal possibility is de1ned as follows: Denition 3.1. (vj : ui | ui ) and (vj : ui | ui ) are called conditional causal possibilities of vj : ui and vj : ui , respectively. In words, conditional causal possibility (vj : ui |ui ) is the possibility that ui causes vj when ui is present. Since (e | e) O = 0 and (eO | e) O = 1 for any event e, (vj : ui | ui ) = 0 and (vj : ui | ui ) = 1 are easily derived from formulae (2) and (3). The probabilistic version P(vj : ui | ui ) of the conditional causal possibility was 1rst introduced by Peng and Reggia [17,18] and is called conditional causal probability. The same notion called causal power was also studied in the psychological literature [1,11], where it is shown that the causal power pij (= P(vj : ui | ui )) is derived by pij = {P(vj | ui ) − P(vj | ui )}\{1 − P(vj | ui )}, if ui is independent of one another. Denition 3.2. I/ the next inequality holds, ui and vj have causality. (vj : ui | ui ) ¿ 0:
(15)
From the de1nition, (vj : ui | ui ) = 0 if ui and vj does not have causality. (vj : ui | ui ) = 1, because (vj : ui | ui ) ∨ (vj : ui | ui ) = 1. Furthermore, (vj : ui ) = 0 is derived using formulae (2) and (10). Thus, the causation event vj : u never happens, if ui and vj does not have causality. Denition 3.3. Let ij be a conjunction of any cause and causation events or their negations other than vj : ui and vj : ui . Then, ij is said to be a context of vj : ui , and also of vj : ui .
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
Denition 3.4. Let ij be a context of vj : ui . Then, i/ the following two equations hold, vj : ui (vj : ui ) is possibilistically causation independent. (vj : ui | ui ∧ ij ) = (vj : ui | ui );
(16a)
(vj : ui | ui ∧ ij ) = (vj : ui | ui );
(16b)
where ui ∧ ij must not be contradictory. De1nition 3.4 shows that values of (vj : ui | ui ) and (vj : ui | ui ) are not a/ected by the states of causes and causation events except for ui and vj : ui , if vj : ui (vj : ui ) is possibilistically causation independent. Note that both (vj : ui | ui ∧ ij ) = (vj : ui |ui ) = 0 and (vj : ui | ui ∧ ij ) = (vj : ui | ui ) = 1 hold even if vj : ui is not possibilistically causation independent. Now, we assume the following: Assumption 3.1. (1) Any ui is possibilistically independent of xi ∈ {ui ; ui }, i = i. The same holds in relation to ui and xi . (2) Any vj : ui and vj : ui are possibilistically causation independent. From the assumptions, Proposition 3.1 is derived. Proposition 3.1. (1) vj : ui is possibilistically independent of xi ∈ {ui ; ui }, i = i. The same holds in relation to vj : ui and xi . (2) Let be a conjunction of ui or ui , or tautology. It must not be contradictory. Then vj : ui (vj : ui ) and vj : ui (vj : ui ) are noninteractive including cases where i = i , when they are conditioned by . Namely, (vj : ui ∧ vj : ui | ) = (vj : ui | ) ∧ (vj : ui | );
(17a)
(vj : ui ∧ vj : ui | ) = (vj : ui | ) ∧ (vj : ui | );
(17b)
(vj : ui ∧ vj : ui | ) = (vj : ui | ) ∧ (vj : ui | ):
(17c)
Proof. See Appendix A. Possibilities of vj and vj can be calculated from (ui ), (ui ), (vj : ui | ui ) and (vj : ui | ui ) using the next proposition. Proposition 3.2. When (ui ), (ui ), (vj : ui | ui ) and (vj : ui | ui ) are given, (vi ) and (vi ) are obtained in the following equations. (vj ) =
((vj : ui | ui ) ∧ (ui ));
(18a)
{((vj : ui | ui ) ∧ (ui )) ∨ (ui )}:
(18b)
i
(vj ) =
i
Proof. See Appendix B.
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
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Then, the next proposition gives a way to calculate possibilities of vj , vj conditioned by given states of the causes. Proposition 3.3. Let ui∗ denote ui or ui , and vj∗ , vj or vj . U = {u1 ; : : : ; uI } and V = {v1 ; : : : ; vJ }. Then, the following equations hold. vj ui∗ = (vj : ui | ui∗ ) ∨ ((vj : ui | ui ) ∧ (ui )); (19a) i∈suf (A) i∈suf (U −A) i∈suf (A) A=∅; A⊆U vj ui∗ = (vj : ui | ui∗ ) ∧ {((vj : ui | ui ) ∧ (ui )) ∨ (ui )}; i∈suf (A) i∈suf (U −A) i∈suf (A) A=∅; A⊆U
(19b)
where suf (A) is a set of su:xes of all elements in set A. Proof. See Appendix C. Finally, the joint possibility of states of e/ects conditioned by given states of causes can be obtained using the next proposition. Proposition 3.4. Let ui∗ denote ui or ui , and vj∗ , vj or vj . U = {u1 ; : : : ; uI } and V = {v1 ; : : : ; vJ }. Then, the next equation holds. (20) vj∗ ui∗ = vj∗ ui∗ : j∈suf (B) i∈suf (A) i∈suf (A) j∈suf (B) A=∅; A⊆U B=∅; B⊆V Proof. See Appendix D. The above propositions are used to solve the possibilistic causality consistency problem discussed in the following section.
4. Possibilistic causality consistency problem The possibilistic causality consistency problem is de1ned in the following [21]. 4.1. Possibilistic causality consistency problem There are a set U of possible causes ui , and a set V of their possible e/ects vj . U and V are disjoint. U ∗ ⊆ U and V ∗ ⊆ V are sets of causes and e/ects whose states are known to be present or absent without any uncertainty. Let us denote the known states by ui∗ and vj∗ . ui∗ denotes ui or ui , and vj∗ does vj or vj . The
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
sets of ui∗ and vj∗ are expressed by P ∗ and Q∗ , respectively. Both sets do not include contradictory states. P ∗ ∪ Q∗ = ∅ is assumed. The states of events in U − U ∗ and V − V ∗ are completely unknown. Then suppose that we have a hypothesis for events in Uˆ ⊆ U −U ∗ and Vˆ ⊆ V −V ∗ , where some are assumed to be present and the others to be absent. The states of events in the hypothesis are denoted by uˆi and vˆj . uˆi ˆ respectively. Pˆ ∪ Qˆ = ∅. denotes ui or ui , and vˆj does vj or vj . The sets of uˆi and vˆj are denoted by Pˆ and Q, ˆ Qˆ | P ∗ ; Q∗ ) and Now, the possibilistic causality consistency problem (PCC problem) is to calculate Pos(P; ˆ Qˆ | P ∗ ; Q∗ ) de1ned by the following, which are conditional possibilities that the hypothesis and its Pos(P; negation are correct given P ∗ ∪ Q∗ , respectively. ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P; (21) uˆi ∧ vˆj ui∗ ∧ vj∗ ; ˆ ˆ i∈suf (P ∗ ) j∈suf (Q∗ ) i∈suf (P) j∈suf (Q) ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P;
uˆi ∧
ˆ j∈suf (Q)
ˆ i∈suf (P)
vˆj ui∗ ∧ vj∗ ; i∈suf (P∗ ) ∗ j∈suf (Q )
(22)
where if P ∗ , Q∗ , Pˆ and=or Qˆ are empty, the conjunctions given by the sets are ignored. (ui ), (ui ), (vj : ui | ui ) and (vj : ui | ui ) are given as a priori knowledge. The PCC problem is an expansion of the problem handled in [19] and also in [17,18], if the di/erence between probability and possibility is ignored. The theories in those papers are based on the AVC model, and the objective is to solve an inverse problem of causalities similarly to studies based on other causal models [9,23]. The inverse problem mentioned here means one to calculate the conditional probability or possibility that a set of causes are present and the others are absent given some observed e/ects. It is equivalent to the PCC problem with P ∗ = Qˆ = ∅ and Pˆ = U . The advantage of handling the PCC problem instead of the inverse one is in the applicability. The framework of the PCC problem would be applicable to many analytical problems with uncertainty, because humans tend to think that they understand things when they have built a causal model among events in their mind. In the models, the observed are not limited to e/ect events and include some of cause events. The hypotheses that they want to evaluate may include states of some unknown e/ects as well as causes. Even in diagnostic problems, such situations are never rare in the real world. In what follows the solution of the PCC problem is derived. First, the following equation holds due to Eqs. (10) and (21). ˆ P ∗ ; Q∗ ); ˆ Qˆ | P ∗ ; Q∗ ) ∧ Pos(P ∗ ; Q∗ ) = Pos(P; ˆ Q; Pos(P;
(23)
ˆ are possibilities that all states in P ∗ ∪ Q∗ and P ∗ ∪ Q∗ ∪ Pˆ ∪ Qˆ ˆ Q) where Pos(P ∗ ; Q∗ ) and Pos(P ∗ ; Q∗ ; P; happen, respectively. They are given by the following equations. Pos(P ∗ ; Q∗ ) = (24) vj∗ ui∗ ∧ ui∗ : ∗ ∗ ∗ j∈suf (Q ) i∈suf (P ) i∈suf (P ) ˆ = ˆ Q) Pos(P ∗ ; Q∗ ; P;
j∈suf (Q∗ )
vj∗ ∧
ˆ j∈suf (Q)
vˆj ui∗ ∧ uˆi ∧ ui∗ ∧ uˆi : i∈suf (P∗ ) ∗) ˆ ˆ i∈suf (P i∈suf (P) i∈suf (P) (25)
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
41
By utilizing Proposition 3.4 and the possibilistic independence of causes, the above two are re-written as follows: Pos(P ∗ ; Q∗ ) = vj∗ ui∗ ∧ (ui∗ ): (26) i∈suf (P∗ ) j∈suf (Q∗ ) i∈suf (P ∗ )
ˆ = ˆ Q) Pos(P ∗ ; Q∗ ; P;
j∈suf (Q∗ )
vj∗ ui∗ ∧ uˆi ∧ vˆj ui∗ ∧ uˆi i∈suf (P∗ ) ∗) ˆ ˆ ˆ i∈suf (P i∈suf (P) i∈suf (P) j∈suf (Q)
∧
(ui∗ ) ∧
i∈suf (P ∗ )
(uˆi ):
(27)
ˆ i∈suf (P)
If P ∗ and=or Pˆ are empty and there are no conditions of the conditional possibilities in the above equations, they are replaced by (vj∗ ) and=or (vˆj ). ˆ P ∗ ; Q∗ ) can be calculated using Proposition 3.3, since (ui ), ˆ Q; Now, the values of Pos(P ∗ ; Q∗ ) and Pos(P; (ui ), (vj : ui | ui ) and (vj : ui | ui ) are given. Then, by applying the principle of the least speci1c solution to Eq. (23), the solution of PCC problem is obtained as follows: ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P;
ˆ ˆ ˆ Q); ˆ Q); if Pos(P ∗ ; Q∗ ) ¿ Pos(P ∗ ; Q∗ ; P; Pos(P ∗ ; Q∗ ; P; ˆ ˆ Q): if Pos(P ∗ ; Q∗ ) = Pos(P ∗ ; Q∗ ; P;
1;
(28)
ˆ Qˆ | P ∗ ; Q∗ ) is calculated by the next equation. Furthermore, Pos(P; ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P;
ˆ i∈suf (P)
uˆi ∨
ˆ j∈suf (Q)
vˆj ui∗ ∧ vj∗ i∈suf (P∗ ) ∗ j∈suf (Q )
= uˆi ui∗ ∧ vj∗ ∨ vˆj ui∗ ∧ vj∗ ; ∗ ∗ ∗ ∗ ˆ ˆ i∈suf (P ) j∈suf (Q ) i∈suf (P ) j∈suf (Q ) i∈suf (P) j∈suf (Q)
(29) ˆ Qˆ | P ∗ ; Q∗ ). where each term in Eq. (29) can be obtained in the same way as that to calculate Pos(P;
5. Numerical example Let us solve a simple diagnostic problem as a numerical example. Suppose that U = {hay-fever (hay);
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
follows: (nc : hay | hay) = 1:0;
(nc : hay | hay) = 0:3;
(high : hay | hay) = 0:2;
(high : hay | hay) = 1:0;
(diarrhea : hay | hay) = 0:0; (nc :
(nc :
(high :
(high :
(diarrhea :
(diarrhea : hay | hay) = 1:0;
(diarrhea :
(nc : fp | fp) = 1:0; (high : fp | fp) = 1:0;
(diarrhea : fp | fp) = 1:0;
(diarrhea : fp | fp) = 0:1:
Now, a patient complains of high-fever. The patient also says that he is su/ering from hay fever. In this case, the possibility of a hypothesis that the patient has Ju and is su/ering from nasal-congestion but not from diarrhea is calculated in the following way. The conditions of the patient is given by P ∗ = {hay} and Q∗ = {high}, and the hypothesis is denoted by Pˆ = {
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
43
Then, the possibility of negation of the hypothesis is calculated. From Eq. (29), the following equation is derived. ˆ Qˆ | P ∗ ; Q∗ ) = (
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
Appendix A. Proof of Proposition 3.1 (1) Using Eqs. (9), (12) and Assumption 3.1, (vj : ui | ui ) = (vj : ui ) is proved as follows:
(vj : ui | ui ) =
(vj : ui ∧ xi | ui ) =
xi ∈{ui ;ui }
=
{(vj : ui | xi ∧ ui ) ∧ (xi | ui )}
xi ∈{ui ;ui }
{(vj : ui | xi ) ∧ (xi )} = (vj : ui ):
xi ∈{ui ;ui }
(vj : ui | ui ) = (vj : ui ), (vj : ui | ui ) = (vj : ui ) and (vj : ui | ui ) = (vj : ui ) are also proved in the same way. (2) Let zij be vj : ui or vj : ui . Then, the next equation is derived from Eqs. (9), (12) and Assumption 3.1.
(zij ∧ zi j | ) =
{(zij ∧ zi j ∧ xi ∧ xi | )}
xi ∈{ui ;ui } xi ∈{ui ;ui }
=
{(zij | zi j ∧ xi ∧ xi ∧ ) ∧ (zi j | xi ∧ xi ∧ ) ∧ (xi | xi ∧ ) ∧ (xi | )}
xi ; x i
=
{(zij | xi ∧ ) ∧ (zi j | xi ∧ ) ∧ (xi | ) ∧ (xi | )}
xi ; x i
=
{(zij ∧ xi | ) ∧ (zi j ∧ xi | )}
xi ;xi
= (zij | ) ∧ (zi j | ):
Appendix B. Proof of Proposition 3.2 From formulae (5), (vj ) =
(vj : ui )
=
i
(vj : ui )
i
is derived. Then, formulae (2) and (10) derive the following. (vj : ui ) = (vj : ui ∧ ui ) = (vj : ui | ui ) ∧ (ui ): Thus, Eq. (18a) is proved. Using formula (7) and Proposition 3.1, the following is obtained. (vj ) =
i
(vj : ui )
=
i
(vj : ui ):
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
45
Furthermore, from Eqs. (10) and (12), the next equation is derived.
(vj : ui ) =
(vj : ui ∧ xi ) =
xi ∈{ui ; ui }
{(vj : ui | xi ) ∧ (xi )}
xi ∈{ui ; ui }
= {(vj : ui | ui ) ∧ (ui )} ∨ (ui ): Thus, Eq. (18b) is proved.
Appendix C. Proof of Proposition 3.3 Let be the conditional part of the left-hand side of Eqs. (19a) and (19b). Then the next equation holds due to formulae (5). (vj | ) =
(vj : ui ) |
=
i
(vj : ui | ):
i
The next equation also holds because of Eqs. (7) and (17c). (vj | ) =
(vj : ui ) |
=
i
(vj : ui | ):
i
Now, the following equations are derived from the possibilistic causation independence of causation events. (vj : ui | ) = (vj : ui | ui );
if i ∈ suf(A):
(vj : ui | ) = (vj : ui | ui );
if i ∈ suf(A):
From Proposition 3.1 and formulae (2), (9), (10), (vj : ui | ) = (vj : ui ) = (vj : ui | ui ) ∧ (ui );
if i ∈= suf (A)
and (vj : ui | ) = (vj : ui ) = {(vj : ui | xi ) ∧ (xi )};
if i ∈= suf (A)
xi ∈{ui ;ui }
are derived. Thus, Eqs. (19a) and (19b) are proved.
Appendix D. Proof of Proposition 3.4 Let be the conditional part of the left-hand side of Eq. (20). Then, if (vj∗1 ∧ vj∗2 | ) = (vj∗1 | ) ∧ (vj∗2 | ) is proved, the proposition is also proved without loss of generality.
46
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
(1) When vj∗1 = vj1 , vj∗2 = vj2 , the following holds because of formulae (5). (vj1 ∧ vj2 | ) =
(vj1 : ui ) ∧
i
=
=
(vj2 : uh ) |
h
(vj1 : ui ∧ vj2 : uh ) |
i; h
(vj1 : ui ∧ vj2 : uh | ):
i; h
Then, using Proposition 3.1 and formula (5), the proposition is proved as follows:
(vj1 : ui ∧ vj2 : uh | ) =
i;h
{(vj1 : ui | ) ∧ (vj2 : uh | )}
i;h
=
i
=
(vj1 : ui | ) ∧
vj 1 : u i |
(vj2 : uh | )
h
∧
i
vj 2 : u h |
h
= (vj1 | ) ∧ (vj2 | ): (2) When vj∗1 = vj1 , vj∗2 = vj2 , formulae (5) and (7) enable the following transformation. (vj1 ∧ vj2 | ) =
(vj1 : ui ) ∧
i
=
(vj2 : uh ) |
h
{(vj1 : uk ) ∧
i
(vj2 : uh )} |
=
(vj1 : ui ) ∧
i
h
Then, Proposition 3.1 and formulae (5) and (7) derive the following:
(vj1 : ui ) ∧
i
=
(vj2 : uh ) |
h
(vj1 : ui | ) ∧
i
=
=
(vj2 : uh | )
h
i
(vj1 : ui | ) ∧
vj 1 : u i |
(vj2 : uh | )
h
∧
i
= (vj1 | ) ∧ (vj2 | ):
h
vj 2 : u h |
h
(vj2 : rh ) | :
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
47
(3) When vj∗1 = vj1 ; vj∗2 = vj2 , the next equation is proved by Eq. (7) and Proposition 3.1. (vj1 ∧ vj2 | ) = (vj1 : ui ) ∧ (vj2 : uh ) | i
=
i
=
h
(vj1 : ui | ) ∧
(vj1 : ui ) |
(vj2 : uh | )
h
∧
i
(vj2 : uh ) |
h
= (vj1 | ) ∧ (vj2 | ):
References [1] P.W. Cheng, From covariation to causation: a causal power theory, Psychol. Rev. 104 (1997) 367–405. [2] G.F. Cooper, The computational complexity of probabilistic inference using Bayesian belief networks, Arti1cial Intelligence 42 (1990) 393–405. [3] P. Dagum, M. Luby, Approximating probabilistic inference in Bayesian belief networks is NP-hard, Arti1cial Intelligence 60 (1993) 141–153. [4] D. Dubois, M. Grabisch, O. De Mouzon, H. Prade, A possibilistic framework for single-fault causal diagnosis under uncertainty, Internat. J. Gen. Systems 30 (2) (2001) 167–192. [5] D. Dubois, H. Prade, Possibilistic inference under matrix form, in: H. Prade, C.V. Negoita (Eds.), Fuzzy Logic in Knowledge S Rheinland, KSoln, 1986, pp. 112–126. Engineering, Verlag, TUT [6] D. Dubois, H. Prade, Possibility Theory, Plenum Press, New York, 1988. [7] D. Dubois, H. Prade, Epistemic entrenchment and possibilistic logic, Arti1cial Intelligence 50 (1991) 223–239. [8] D. Dubois, H. Prade, A survey of belief revision and updating rules in various uncertainty models, Internat. J. Intelligent Systems 9 (1994) 61–100. [9] D. Dubois, H. Prade, Fuzzy relation equations and causal reasoning, Fuzzy Sets and Systems 75 (1995) 119–134. [10] D. Dubois, H. Prade, A synthetic view of belief revision with uncertain inputs in the framework of possibility theory, Internat. J. Approx. Reason. 17 (1997) 295–324. [11] C. Glymour, Learning causes: psychological explanations of causal explanation, Minds and Machines 8 (1998) 39–60. [12] D. Heckerman, C. Meek, Models and selection criteria for regression and classi1cation, Proc. 13th Conf. on Uncertainty in AI, Providence, 1997, pp. 223–228. [13] E. Hisdal, Conditional possibilities independence and non-interaction, Fuzzy Sets and Systems 1 (1978) 283–297. [14] J.R. Nail, C.S. Wallace, K.B. Korb, Learning Bayesian networks with restricted causal interactions, Proc. 15th Conf. on Uncertainty in AI, Stockholm, 1999, pp. 486 – 493. [15] J. Pearl, Fusion, propagation, and structuring in belief networks, Arti1cial Intelligence 29 (1986) 241–288. [16] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann, London, 1988. [17] Y. Peng, J.A. Reggia, A probabilistic causal model for diagnostic problem solving—Part I: integrating symbolic causal inference with numeric probabilistic inference, IEEE Trans. Systems, Man, Cybernet. SME-17 (2) (1987) 146–162. [18] Y. Peng, J.A. Reggia, Abductive Inference Models for Diagnostic Problem-Solving, Springer, Berlin, 1990. [19] K. Yamada, Conditional causal possibility and its application to inverse problem of causalities, J. Japan Soc. Fuzzy Theory Systems 11 (1999) 104–111 (in Japanese). [20] K. Yamada, Causal reasoning using conditional causal possibilities to express uncertainty of causalities, IFSA World Congress 99, Taipei (2) (1999) 665–669. [21] K. Yamada, Causality consistency problem for two di/erent possibilistic causal models, Internat. Symp. on Medical Informatics and Fuzzy Technology, 1999, Hanoi, pp. 182–189.
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[22] K. Yamada, Possibilistic causality consistency problem with compound causal relationships, The 4th Asian Fuzzy Systems Symposium, Tsukuba, 2000, pp. 89 –94. [23] K. Yamada, M. Honda, A method of diagnosis using possibility theory, Proc. IEA-AIE 96, Fukuoka (1996) 149–154. [24] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3–28.
www.elsevier.com/locate/fss
Possibilistic causality consistency problem based on asymmetrically-valued causal model Koichi Yamada ∗ Department of Management and Information Systems Science, Nagaoka University of Technology, 1603-1 Kami-tomioka, Nagaoka, Niigata 920-2188, Japan Received 11 January 2000; received in revised form 20 February 2002; accepted 22 February 2002
Abstract 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 e/ects. Uncertainties of the causalities are given by conditional causal possibilities, which were proposed to express the exact degrees of possibility of causalities. The expression of the uncertainty also has 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. The paper starts with the de1nition of a causal model called asymmetrically-valued causal model. The conditional causal possibilities are de1ned on the causal model, and their mathematical properties are discussed. Then, the paper de1nes the possibilistic causality consistency problem based on the proposed model and shows how to solve the problem. The discussed problem is one to calculate the conditional possibility of a hypothesis about presence and absence of c 2002 Elsevier Science B.V. All rights reserved. unknown events when the states of some other events are known. Keywords: Possibility theory; Causal reasoning; Causal network; Uncertain reasoning
1. Introduction Modeling causal relationships among events, and predicting or diagnosing unknown events from the known are very common tasks in systems analysis. The language ordinarily used to model the causalities in the 1eld of engineering is numerical formulae. However, it is also popular to model causalities on causal networks, when the objective system is complex and di:cult to model with numerical formulae. Probabilistic causal networks could be e/ective, especially when the causalities are uncertain. Belief networks (BNs), which are a type of well-established probabilistic networks, are attracting a great deal of attention for this reason [15,16]. However, it is reported that their application is limited [14] in spite of the popularity of the theoretical research due to the fact that problems of learning BNs and probabilistic inference using them are NP-hard [2,3,14], while the recent research is focusing on reduction of the complexity [12,14]. BNs also have a problem to need a huge number of conditional probabilities as a priori knowledge; as many as combinations of states of all possible causes for each e/ect. ∗
Tel.=fax: +81-258-47-9359. E-mail address: [email protected] (K. Yamada).
c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/02/$ - see front matter PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 0 9 9 - 4
34
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
A possible alternative of the ordinary BN is a probabilistic causal model based on conditional causal probabilities proposed by [17,18], if the objective system can be modeled by the “noisy-or-gate” where causes generate e/ects disjunctively [16]. In fact, it seems that many diagnostic problems in industry can be modeled by the noisy-or-gate, considering that a more restrictive assumption such as single fault assumption has been common in the 1eld. One of the advantages of the model is that it needs only as many conditional causal probabilities as the number of possible causes for each e/ect. The conditional causal probabilities are those devised to express the exact probabilities of causalities and the same as causal powers studied in psychological literature [1,11]. One of the reasons why they are studied in the 1eld of psychology might be that they reJect the probabilities of causalities that a human recognizes in her mind more accurately than conditional probabilities [1,18]. Thus the use of conditional causal probabilities is more appropriate than the conditional probabilities when we cannot help giving the subjective probabilities instead of statistical ones [18]. Now, probability is a ratio scale of uncertainty. 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 di/erence in prior probabilities may a/ect the diagnostic results, namely the probabilistic orders of possible causes [23]. There is another scale of uncertainty called possibility [6,13,24]. Since possibility is essentially an ordinal scale [7,8], reasoning based on the theory is insensitive to some errors of evaluation of uncertainty. Thus it might be suitable for problems with not enough data to calculate reliable probabilities. Possibilistic causal reasoning has recently been studied in this context [9,23,19,20,21]. In early days of these studies, conditional possibilities were used to express uncertainty of causalities, as conditional probabilities are [23]. Nowadays, however, approaches with conditional causal possibilities, which are a possibilistic version of conditional causal probabilities, have been proposed [19,20,21]. 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 [19,20]. Dubois and Prade also proposed a model of causal reasoning with substantially the same representation of uncertainty of causalities in [9]. The model is expressive for uncertainty in the sense that it handles uncertainty of observation as well as of causalities with two types of representation: “more or less certainly true” and “more or less certainly false” (equivalently, “more or less certainly true” and “more or less possibly true”). Then, the model was expanded in [4], by combining binary states of e/ects with fuzzy sets on numerical attributes. In those models, however, the available observation is limited to e/ects (manifestations) and the hypothesis is to causes (disorders). The paper addresses uncertain reasoning on a two-layered causal network with the conditional causal possibilities. More speci1cally, it de1nes the asymmetrically-valued causal model, a type of causal model with causation events used in [19,21]. Then, it de1nes and solves the possibilistic causality consistency problem based on the proposed model. It is a problem of calculating the conditional possibility of a hypothesis about presence=absence of unknown events, when the states of some other events are known [21]. This problem includes, as special cases, an inverse problem of causalities which calculates the conditional possibility that a set of causes are present and the others are absent given some observed e/ects [23,19]. It also includes a causality analysis problem calculating the conditional possibility of presence=absence of each unknown event when the states of some other events are known [20]. BrieJy, the paper is an expanded one of [19] and [21] in the sense that (1) the causality consistency problem includes the inverse problem of causalities, (2) the possibility of negation of the hypothesis is calculated as well as the possibility of the hypothesis, and (3) some proofs in the papers are re1ned mathematically to be correct.
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
35
2. Asymmetrically-valued causal model The section de1nes cause, e/ect, and causation events proposed by [17,18] and the asymmetrically-valued causal model introduced in [21]. Denition 2.1. Let U = {ui | i = 1; : : : ; I } and V = {vj | j = 1; : : : ; J } be two sets of primitive events. U and V are disjoint. The presence of vj is dependent on at least one of ui . Thus, ui and vj are the cause and e4ect (events), respectively. The presence and absence of events are described by classical propositions; a proposition representing presence of an event is denoted by the event itself (e.g. ui , vj ), and one representing absence is by its negation (e.g. ui , vj ). The single symbol vj : ui denotes a causation event (or its presence) that means “ui arises, and ui actually causes vj ”. The causation events satisfy the next logical formula: vj : ui → ui ∧ vj ;
(1)
where ∧ and → are conjunction and implication, respectively. The set of all possible causation events is denoted by V : U = {vj : ui | vj ∈ V; ui ∈ U }. Then, following formulae are easily derived from formula (1). vj : ui ↔ (vj : ui ) ∧ ui ↔ (vj : ui ) ∧ vj ↔ (vj : ui ) ∧ ui ∧ vj ;
(2)
vj : u i ∧ ui ↔ ui ;
(3)
vj : u i ∧ vj ↔ vj ;
(4)
where ↔ denotes equivalence. Peng and Reggia state that the occurrence of causation event vj : ui represents the realization or instantiation of the causal link between ui and vj [18]. They also explain that it is a distinct type of basic event that cannot be expressed by a logical expression of events ui and vj . It is understandable considering the situation, where vj : ui is false even if ui ∧ vj is true. The situation arises if ui and vj are present, but vj is caused by some other cause. Then the asymmetrically-valued causal model is de1ned using the causation events. Denition 2.2. Let Ui = {ui ; ui } and Vj = {vj ; vj }. If the following formula is satis1ed, ui (or Ui ) and vj (or Vj ) follow the asymmetrically-valued causal model (AVC model). vj ↔ (vj : ui ); (5) i
where ∨ denotes disjunction. The above de1nition, together with formula (1), derives the next formula. vj → ui :
(6)
i
Thus, if some e/ect is present, some cause must be present. The AVC model is used in most inverse problems of causalities employing causation events [17,18,19], and formula (5) is called the mandatory causation assumption. If Ui = {ui1 ; : : : ; uin }, Vj = {vj1 ; : : : ; vjm } and vjk ↔ i; h (vjk : uih ) as in [20], uih (or Ui ) and vjk (or Vj ) follow the symmetrically-valued causal (SVC) model [21]. In the case of the SVC model, any element in Vj cannot be present unless it is caused by a cause or
36
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
some causes. For example, suppose that Vj is a set of possible skin colors of chameleon j and that Ui is a set of possible colors of object i around the chameleons. The positions of the chameleons and the objects are supposed to be 1xed. If the color of a chameleon j is always caused by a color or some colors of the objects in a disjunctive way, Ui and Vj follows the SVC model. Note that there can exist a binary SVC model: the model with Ui = {ui1 ; ui2 }, Vj = {vj1 ; vj2 } and vjk ↔ i; h (vjk : uih ), (h; k = 1; 2). In this case, ifui1 and ui2 (vj1 and vj2 ) are regarded as ui and ui (vj and vj ) respectively, vj ↔ i {(vj : ui ) ∨ (vj : ui )} and vj ↔ i {(vj : ui ) ∨ (vj : ui )} hold. This means that absence of ui may cause vj and that absence of vj may be caused by presence or absence of ui . In the AVC model, however, vj is never caused by any ui as known from the mandatory causation assumption. In addition, vj is not caused by any cause or its negation di/erently from the SVC, and the following formula holds. vj ↔ (vj : ui ): (7) i
In the rest of this paper, we assume Ui = {ui ; ui } and Vj = {vj ; vj }, and that they follow the AVC model. 3. Conditional causal possibility Possibility is one of the scales for measuring uncertainty [6,24]. Let (s i ), i = 1; : : : ; n be a possibility distribution on a universal set S = {s1 ; : : : ; sn }. The distribution must satisfy i (si ) = 1, where ∨(∧) denotes max (min) when used for possibilities, while it denotes disjunction (conjunction) for propositions in the paper. Then, the possibility measure (A), A ⊆ S could be de1ned using the distribution. (si ); (8) (A) = si ∈A
where (A) = 0 if A is the empty set. In Eq. (8), the argument of (•) is a subset of the universe S. However, in the paper, notation such as ( si ∈A si ) is used instead in order to handle possibility of events denoted by logical expressions. Thus, possibility of an event corresponding to a subset with a single element si is denoted by (si ) instead of ({si }). Namely, (si ) = (si ) holds in the notation. In the rest of this paper, possibility of si is denoted by (si ) for convenience. Let ek (k = 1; 2; 3; : : :) be an event representing ui or vj : u. Both of them are elementary events in the sense that they cannot be de1ned by any logical expression of the other events. vj is a composite one, because it can be de1ned by formula (5). Then, let ek∗ be ek or ek , and (ek∗ ) be a possibility that is equal to a marginal possibility distribution (ek∗ ) on Ek = {ek ; ek }. Note that the distributions are de1ned on {ui ; ui } or {vj : ui ; vj : ui } in the paper, but not on U nor V : U . Due to formulae (2) – (4), (vj : ui )6(vj ), (vj : ui )¿(ui ) and (vj : ui )¿(vj ) must be satis1ed. Let (e1∗ ∧ e2∗ ) be a possibility of the joint event e1∗ ; e2∗ , and be equal to the joint possibility distribution (e1∗ ; e2∗ ) on E1 × E. Then, possibility (e1∗ ) is equal to the maximum of (e1∗ ∧ e2 ) and (e1∗ ∧ e2 ). (e1∗ ∧ x): (9) (e1∗ ) = x∈{e2 ;e2 }
For conditional possibility, it is assumed that (d | e) ∨ (dO | e) = 1 holds for any events e and d. The relation between joint and conditional possibilities is given by (e1∗ ∧ e2∗ ) = (e1∗ | e2∗ ) ∧ (e2∗ );
(10)
which was 1rst introduced in [13] as the equation corresponding to the multiplicative law of probability.
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
37
Suppose that (e1∗ ∧ e2∗ ) and (e2∗ ) are known and (e1∗ | e2∗ ) is unknown in Eq. (10). Though the solution is not unique in this case, the paper chooses the largest one as its solution according to the idea of the principle of the maximal speci1city or the least speci1c solution [5,8,10]. In the case of Eq. (10), the principle gives the next solution. (e1∗ ∧ e2∗ ); if (e2∗ ) ¿ (e1∗ ∧ e2∗ ); ∗ ∗ (e1 | e2 ) = (11) 1; if (e2∗ ) = (e1∗ ∧ e2∗ ): Then, the next equality is easily obtained using Eq. (11). (e1∗ ∧ e2∗ | e3∗ ) = (e1∗ | e2∗ ∧ e3∗ ) ∧ (e2∗ | e3∗ ):
(12)
Now, if Eq. (13) holds, it is said that (e2∗ ) is possibilistically independent of e1∗ [13]. When Eq. (14) is satis1ed, e1∗ and e2∗ are noninteractive [24]. (e2∗ | e1∗ ) = (e2∗ );
(13)
(e1∗ ∧ e2∗ ) = (e1∗ ): ∧ (e2∗ ):
(14)
e1∗ and e2∗ are noninteractive, if e2∗ (e1∗ ) is possibilistically independent of e1∗ (e2∗ ). For the correspondence between the logical and the possibilistic expression, it is de1ned that any event e is true (present) i/ (e) = 1 and (e) O = 0. I/ (e) = 0 and (e) O = 1, e is false (absent). Then, it is easily obtained that (e | e) = (eO | e) O = 1 by substituting e for e1∗ and e2∗ in Eq. (11). Furthermore, (e | e) O =0 holds except the case where (e) O = 0, and (eO | e) = 0 when (e) = 0. Now, conditional causal possibility is de1ned as follows: Denition 3.1. (vj : ui | ui ) and (vj : ui | ui ) are called conditional causal possibilities of vj : ui and vj : ui , respectively. In words, conditional causal possibility (vj : ui |ui ) is the possibility that ui causes vj when ui is present. Since (e | e) O = 0 and (eO | e) O = 1 for any event e, (vj : ui | ui ) = 0 and (vj : ui | ui ) = 1 are easily derived from formulae (2) and (3). The probabilistic version P(vj : ui | ui ) of the conditional causal possibility was 1rst introduced by Peng and Reggia [17,18] and is called conditional causal probability. The same notion called causal power was also studied in the psychological literature [1,11], where it is shown that the causal power pij (= P(vj : ui | ui )) is derived by pij = {P(vj | ui ) − P(vj | ui )}\{1 − P(vj | ui )}, if ui is independent of one another. Denition 3.2. I/ the next inequality holds, ui and vj have causality. (vj : ui | ui ) ¿ 0:
(15)
From the de1nition, (vj : ui | ui ) = 0 if ui and vj does not have causality. (vj : ui | ui ) = 1, because (vj : ui | ui ) ∨ (vj : ui | ui ) = 1. Furthermore, (vj : ui ) = 0 is derived using formulae (2) and (10). Thus, the causation event vj : u never happens, if ui and vj does not have causality. Denition 3.3. Let ij be a conjunction of any cause and causation events or their negations other than vj : ui and vj : ui . Then, ij is said to be a context of vj : ui , and also of vj : ui .
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
Denition 3.4. Let ij be a context of vj : ui . Then, i/ the following two equations hold, vj : ui (vj : ui ) is possibilistically causation independent. (vj : ui | ui ∧ ij ) = (vj : ui | ui );
(16a)
(vj : ui | ui ∧ ij ) = (vj : ui | ui );
(16b)
where ui ∧ ij must not be contradictory. De1nition 3.4 shows that values of (vj : ui | ui ) and (vj : ui | ui ) are not a/ected by the states of causes and causation events except for ui and vj : ui , if vj : ui (vj : ui ) is possibilistically causation independent. Note that both (vj : ui | ui ∧ ij ) = (vj : ui |ui ) = 0 and (vj : ui | ui ∧ ij ) = (vj : ui | ui ) = 1 hold even if vj : ui is not possibilistically causation independent. Now, we assume the following: Assumption 3.1. (1) Any ui is possibilistically independent of xi ∈ {ui ; ui }, i = i. The same holds in relation to ui and xi . (2) Any vj : ui and vj : ui are possibilistically causation independent. From the assumptions, Proposition 3.1 is derived. Proposition 3.1. (1) vj : ui is possibilistically independent of xi ∈ {ui ; ui }, i = i. The same holds in relation to vj : ui and xi . (2) Let be a conjunction of ui or ui , or tautology. It must not be contradictory. Then vj : ui (vj : ui ) and vj : ui (vj : ui ) are noninteractive including cases where i = i , when they are conditioned by . Namely, (vj : ui ∧ vj : ui | ) = (vj : ui | ) ∧ (vj : ui | );
(17a)
(vj : ui ∧ vj : ui | ) = (vj : ui | ) ∧ (vj : ui | );
(17b)
(vj : ui ∧ vj : ui | ) = (vj : ui | ) ∧ (vj : ui | ):
(17c)
Proof. See Appendix A. Possibilities of vj and vj can be calculated from (ui ), (ui ), (vj : ui | ui ) and (vj : ui | ui ) using the next proposition. Proposition 3.2. When (ui ), (ui ), (vj : ui | ui ) and (vj : ui | ui ) are given, (vi ) and (vi ) are obtained in the following equations. (vj ) =
((vj : ui | ui ) ∧ (ui ));
(18a)
{((vj : ui | ui ) ∧ (ui )) ∨ (ui )}:
(18b)
i
(vj ) =
i
Proof. See Appendix B.
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
39
Then, the next proposition gives a way to calculate possibilities of vj , vj conditioned by given states of the causes. Proposition 3.3. Let ui∗ denote ui or ui , and vj∗ , vj or vj . U = {u1 ; : : : ; uI } and V = {v1 ; : : : ; vJ }. Then, the following equations hold. vj ui∗ = (vj : ui | ui∗ ) ∨ ((vj : ui | ui ) ∧ (ui )); (19a) i∈suf (A) i∈suf (U −A) i∈suf (A) A=∅; A⊆U vj ui∗ = (vj : ui | ui∗ ) ∧ {((vj : ui | ui ) ∧ (ui )) ∨ (ui )}; i∈suf (A) i∈suf (U −A) i∈suf (A) A=∅; A⊆U
(19b)
where suf (A) is a set of su:xes of all elements in set A. Proof. See Appendix C. Finally, the joint possibility of states of e/ects conditioned by given states of causes can be obtained using the next proposition. Proposition 3.4. Let ui∗ denote ui or ui , and vj∗ , vj or vj . U = {u1 ; : : : ; uI } and V = {v1 ; : : : ; vJ }. Then, the next equation holds. (20) vj∗ ui∗ = vj∗ ui∗ : j∈suf (B) i∈suf (A) i∈suf (A) j∈suf (B) A=∅; A⊆U B=∅; B⊆V Proof. See Appendix D. The above propositions are used to solve the possibilistic causality consistency problem discussed in the following section.
4. Possibilistic causality consistency problem The possibilistic causality consistency problem is de1ned in the following [21]. 4.1. Possibilistic causality consistency problem There are a set U of possible causes ui , and a set V of their possible e/ects vj . U and V are disjoint. U ∗ ⊆ U and V ∗ ⊆ V are sets of causes and e/ects whose states are known to be present or absent without any uncertainty. Let us denote the known states by ui∗ and vj∗ . ui∗ denotes ui or ui , and vj∗ does vj or vj . The
40
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
sets of ui∗ and vj∗ are expressed by P ∗ and Q∗ , respectively. Both sets do not include contradictory states. P ∗ ∪ Q∗ = ∅ is assumed. The states of events in U − U ∗ and V − V ∗ are completely unknown. Then suppose that we have a hypothesis for events in Uˆ ⊆ U −U ∗ and Vˆ ⊆ V −V ∗ , where some are assumed to be present and the others to be absent. The states of events in the hypothesis are denoted by uˆi and vˆj . uˆi ˆ respectively. Pˆ ∪ Qˆ = ∅. denotes ui or ui , and vˆj does vj or vj . The sets of uˆi and vˆj are denoted by Pˆ and Q, ˆ Qˆ | P ∗ ; Q∗ ) and Now, the possibilistic causality consistency problem (PCC problem) is to calculate Pos(P; ˆ Qˆ | P ∗ ; Q∗ ) de1ned by the following, which are conditional possibilities that the hypothesis and its Pos(P; negation are correct given P ∗ ∪ Q∗ , respectively. ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P; (21) uˆi ∧ vˆj ui∗ ∧ vj∗ ; ˆ ˆ i∈suf (P ∗ ) j∈suf (Q∗ ) i∈suf (P) j∈suf (Q) ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P;
uˆi ∧
ˆ j∈suf (Q)
ˆ i∈suf (P)
vˆj ui∗ ∧ vj∗ ; i∈suf (P∗ ) ∗ j∈suf (Q )
(22)
where if P ∗ , Q∗ , Pˆ and=or Qˆ are empty, the conjunctions given by the sets are ignored. (ui ), (ui ), (vj : ui | ui ) and (vj : ui | ui ) are given as a priori knowledge. The PCC problem is an expansion of the problem handled in [19] and also in [17,18], if the di/erence between probability and possibility is ignored. The theories in those papers are based on the AVC model, and the objective is to solve an inverse problem of causalities similarly to studies based on other causal models [9,23]. The inverse problem mentioned here means one to calculate the conditional probability or possibility that a set of causes are present and the others are absent given some observed e/ects. It is equivalent to the PCC problem with P ∗ = Qˆ = ∅ and Pˆ = U . The advantage of handling the PCC problem instead of the inverse one is in the applicability. The framework of the PCC problem would be applicable to many analytical problems with uncertainty, because humans tend to think that they understand things when they have built a causal model among events in their mind. In the models, the observed are not limited to e/ect events and include some of cause events. The hypotheses that they want to evaluate may include states of some unknown e/ects as well as causes. Even in diagnostic problems, such situations are never rare in the real world. In what follows the solution of the PCC problem is derived. First, the following equation holds due to Eqs. (10) and (21). ˆ P ∗ ; Q∗ ); ˆ Qˆ | P ∗ ; Q∗ ) ∧ Pos(P ∗ ; Q∗ ) = Pos(P; ˆ Q; Pos(P;
(23)
ˆ are possibilities that all states in P ∗ ∪ Q∗ and P ∗ ∪ Q∗ ∪ Pˆ ∪ Qˆ ˆ Q) where Pos(P ∗ ; Q∗ ) and Pos(P ∗ ; Q∗ ; P; happen, respectively. They are given by the following equations. Pos(P ∗ ; Q∗ ) = (24) vj∗ ui∗ ∧ ui∗ : ∗ ∗ ∗ j∈suf (Q ) i∈suf (P ) i∈suf (P ) ˆ = ˆ Q) Pos(P ∗ ; Q∗ ; P;
j∈suf (Q∗ )
vj∗ ∧
ˆ j∈suf (Q)
vˆj ui∗ ∧ uˆi ∧ ui∗ ∧ uˆi : i∈suf (P∗ ) ∗) ˆ ˆ i∈suf (P i∈suf (P) i∈suf (P) (25)
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
41
By utilizing Proposition 3.4 and the possibilistic independence of causes, the above two are re-written as follows: Pos(P ∗ ; Q∗ ) = vj∗ ui∗ ∧ (ui∗ ): (26) i∈suf (P∗ ) j∈suf (Q∗ ) i∈suf (P ∗ )
ˆ = ˆ Q) Pos(P ∗ ; Q∗ ; P;
j∈suf (Q∗ )
vj∗ ui∗ ∧ uˆi ∧ vˆj ui∗ ∧ uˆi i∈suf (P∗ ) ∗) ˆ ˆ ˆ i∈suf (P i∈suf (P) i∈suf (P) j∈suf (Q)
∧
(ui∗ ) ∧
i∈suf (P ∗ )
(uˆi ):
(27)
ˆ i∈suf (P)
If P ∗ and=or Pˆ are empty and there are no conditions of the conditional possibilities in the above equations, they are replaced by (vj∗ ) and=or (vˆj ). ˆ P ∗ ; Q∗ ) can be calculated using Proposition 3.3, since (ui ), ˆ Q; Now, the values of Pos(P ∗ ; Q∗ ) and Pos(P; (ui ), (vj : ui | ui ) and (vj : ui | ui ) are given. Then, by applying the principle of the least speci1c solution to Eq. (23), the solution of PCC problem is obtained as follows: ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P;
ˆ ˆ ˆ Q); ˆ Q); if Pos(P ∗ ; Q∗ ) ¿ Pos(P ∗ ; Q∗ ; P; Pos(P ∗ ; Q∗ ; P; ˆ ˆ Q): if Pos(P ∗ ; Q∗ ) = Pos(P ∗ ; Q∗ ; P;
1;
(28)
ˆ Qˆ | P ∗ ; Q∗ ) is calculated by the next equation. Furthermore, Pos(P; ˆ Qˆ | P ∗ ; Q∗ ) = Pos(P;
ˆ i∈suf (P)
uˆi ∨
ˆ j∈suf (Q)
vˆj ui∗ ∧ vj∗ i∈suf (P∗ ) ∗ j∈suf (Q )
= uˆi ui∗ ∧ vj∗ ∨ vˆj ui∗ ∧ vj∗ ; ∗ ∗ ∗ ∗ ˆ ˆ i∈suf (P ) j∈suf (Q ) i∈suf (P ) j∈suf (Q ) i∈suf (P) j∈suf (Q)
(29) ˆ Qˆ | P ∗ ; Q∗ ). where each term in Eq. (29) can be obtained in the same way as that to calculate Pos(P;
5. Numerical example Let us solve a simple diagnostic problem as a numerical example. Suppose that U = {hay-fever (hay);
42
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
follows: (nc : hay | hay) = 1:0;
(nc : hay | hay) = 0:3;
(high : hay | hay) = 0:2;
(high : hay | hay) = 1:0;
(diarrhea : hay | hay) = 0:0; (nc :
(nc :
(high :
(high :
(diarrhea :
(diarrhea : hay | hay) = 1:0;
(diarrhea :
(nc : fp | fp) = 1:0; (high : fp | fp) = 1:0;
(diarrhea : fp | fp) = 1:0;
(diarrhea : fp | fp) = 0:1:
Now, a patient complains of high-fever. The patient also says that he is su/ering from hay fever. In this case, the possibility of a hypothesis that the patient has Ju and is su/ering from nasal-congestion but not from diarrhea is calculated in the following way. The conditions of the patient is given by P ∗ = {hay} and Q∗ = {high}, and the hypothesis is denoted by Pˆ = {
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
43
Then, the possibility of negation of the hypothesis is calculated. From Eq. (29), the following equation is derived. ˆ Qˆ | P ∗ ; Q∗ ) = (
44
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
Appendix A. Proof of Proposition 3.1 (1) Using Eqs. (9), (12) and Assumption 3.1, (vj : ui | ui ) = (vj : ui ) is proved as follows:
(vj : ui | ui ) =
(vj : ui ∧ xi | ui ) =
xi ∈{ui ;ui }
=
{(vj : ui | xi ∧ ui ) ∧ (xi | ui )}
xi ∈{ui ;ui }
{(vj : ui | xi ) ∧ (xi )} = (vj : ui ):
xi ∈{ui ;ui }
(vj : ui | ui ) = (vj : ui ), (vj : ui | ui ) = (vj : ui ) and (vj : ui | ui ) = (vj : ui ) are also proved in the same way. (2) Let zij be vj : ui or vj : ui . Then, the next equation is derived from Eqs. (9), (12) and Assumption 3.1.
(zij ∧ zi j | ) =
{(zij ∧ zi j ∧ xi ∧ xi | )}
xi ∈{ui ;ui } xi ∈{ui ;ui }
=
{(zij | zi j ∧ xi ∧ xi ∧ ) ∧ (zi j | xi ∧ xi ∧ ) ∧ (xi | xi ∧ ) ∧ (xi | )}
xi ; x i
=
{(zij | xi ∧ ) ∧ (zi j | xi ∧ ) ∧ (xi | ) ∧ (xi | )}
xi ; x i
=
{(zij ∧ xi | ) ∧ (zi j ∧ xi | )}
xi ;xi
= (zij | ) ∧ (zi j | ):
Appendix B. Proof of Proposition 3.2 From formulae (5), (vj ) =
(vj : ui )
=
i
(vj : ui )
i
is derived. Then, formulae (2) and (10) derive the following. (vj : ui ) = (vj : ui ∧ ui ) = (vj : ui | ui ) ∧ (ui ): Thus, Eq. (18a) is proved. Using formula (7) and Proposition 3.1, the following is obtained. (vj ) =
i
(vj : ui )
=
i
(vj : ui ):
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
45
Furthermore, from Eqs. (10) and (12), the next equation is derived.
(vj : ui ) =
(vj : ui ∧ xi ) =
xi ∈{ui ; ui }
{(vj : ui | xi ) ∧ (xi )}
xi ∈{ui ; ui }
= {(vj : ui | ui ) ∧ (ui )} ∨ (ui ): Thus, Eq. (18b) is proved.
Appendix C. Proof of Proposition 3.3 Let be the conditional part of the left-hand side of Eqs. (19a) and (19b). Then the next equation holds due to formulae (5). (vj | ) =
(vj : ui ) |
=
i
(vj : ui | ):
i
The next equation also holds because of Eqs. (7) and (17c). (vj | ) =
(vj : ui ) |
=
i
(vj : ui | ):
i
Now, the following equations are derived from the possibilistic causation independence of causation events. (vj : ui | ) = (vj : ui | ui );
if i ∈ suf(A):
(vj : ui | ) = (vj : ui | ui );
if i ∈ suf(A):
From Proposition 3.1 and formulae (2), (9), (10), (vj : ui | ) = (vj : ui ) = (vj : ui | ui ) ∧ (ui );
if i ∈= suf (A)
and (vj : ui | ) = (vj : ui ) = {(vj : ui | xi ) ∧ (xi )};
if i ∈= suf (A)
xi ∈{ui ;ui }
are derived. Thus, Eqs. (19a) and (19b) are proved.
Appendix D. Proof of Proposition 3.4 Let be the conditional part of the left-hand side of Eq. (20). Then, if (vj∗1 ∧ vj∗2 | ) = (vj∗1 | ) ∧ (vj∗2 | ) is proved, the proposition is also proved without loss of generality.
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K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
(1) When vj∗1 = vj1 , vj∗2 = vj2 , the following holds because of formulae (5). (vj1 ∧ vj2 | ) =
(vj1 : ui ) ∧
i
=
=
(vj2 : uh ) |
h
(vj1 : ui ∧ vj2 : uh ) |
i; h
(vj1 : ui ∧ vj2 : uh | ):
i; h
Then, using Proposition 3.1 and formula (5), the proposition is proved as follows:
(vj1 : ui ∧ vj2 : uh | ) =
i;h
{(vj1 : ui | ) ∧ (vj2 : uh | )}
i;h
=
i
=
(vj1 : ui | ) ∧
vj 1 : u i |
(vj2 : uh | )
h
∧
i
vj 2 : u h |
h
= (vj1 | ) ∧ (vj2 | ): (2) When vj∗1 = vj1 , vj∗2 = vj2 , formulae (5) and (7) enable the following transformation. (vj1 ∧ vj2 | ) =
(vj1 : ui ) ∧
i
=
(vj2 : uh ) |
h
{(vj1 : uk ) ∧
i
(vj2 : uh )} |
=
(vj1 : ui ) ∧
i
h
Then, Proposition 3.1 and formulae (5) and (7) derive the following:
(vj1 : ui ) ∧
i
=
(vj2 : uh ) |
h
(vj1 : ui | ) ∧
i
=
=
(vj2 : uh | )
h
i
(vj1 : ui | ) ∧
vj 1 : u i |
(vj2 : uh | )
h
∧
i
= (vj1 | ) ∧ (vj2 | ):
h
vj 2 : u h |
h
(vj2 : rh ) | :
K. Yamada / Fuzzy Sets and Systems 132 (2002) 33 – 48
47
(3) When vj∗1 = vj1 ; vj∗2 = vj2 , the next equation is proved by Eq. (7) and Proposition 3.1. (vj1 ∧ vj2 | ) = (vj1 : ui ) ∧ (vj2 : uh ) | i
=
i
=
h
(vj1 : ui | ) ∧
(vj1 : ui ) |
(vj2 : uh | )
h
∧
i
(vj2 : uh ) |
h
= (vj1 | ) ∧ (vj2 | ):
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