Possibility Theory in the Management of Chance Discovery

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1 From: AAAI Techical Report FS Compilatio copyright 00, AAAI ( All rights reserved. Possibility Theory i the Maagemet of Chace Discovery Roald R. Yager Machie Itelligece Istitute Ioa College New Rochelle, NY yager@paix.com Itroductio The purpose this work is to discuss ad ivestigate the potetial role of possibility theory [1-4] ad other o-stadard ucertaity calculi, such as the Dempster- Shafer theory [5, 6] of evidece, i the developmet of a framework for the maagemet of chace discovery. Possibility theory provides a framework for the represetatio ad maagemet of iformatio about ucertai situatios. It is closely related to fuzzy set theory. Possibility theory uses two dual measures, called the possibility measure ad certaity (ecessity) measure, to model available iformatio about ucertaity situatios. As opposed to probability theory. where the probability measure is additive, the possibility measure ivolves Maxig of its elemets ad the Certaity measure, which is the dual of the possibility ivolves Miig of its elemets. I additio possibility theory is ot as strogly costraied by the law of the excluded middle allowig a more flexible framework. A particularly importat feature of possibility theory ad these other o-stadard calculi are their ability to represet iformatio that is ot impossible. Here we have a framework i which we are able to maage outcomes that are ot expressly deied although ever have occurred. Its close relatioship with fuzzy set theory makes it very suitable for the represetatio of iformatio about ucertaity that is ot ecessarily of a radom type. Here we shall focus o the issue of decisio i the face of possibility ad other types of ucertaity, The basic issues ivolved i decisio makig uder icertitude ca best be uderstood usig the decisio matrix show i figure #1. I this matrix the Ai correspod to a collectio of alterative courses of actio ope to a decisio maker, oe of these actios must be selected by the decisio maker. The elemets x j are possible values associated with a variable V. Geerally the value of V is ukow before the decisio maker must make a selectio from amog the alterative courses of actio. Our kowledge about V is captured by our ucertaity calculi. Fially C ij is the payoff to the decisio maker if he selects alterative Ai ad V assumes the value X j. Oe approach for selectig the best alterative i this decisio problem is to calculate for each alterative a value Val(A i ) ad the select the alterative that has the largest of these values. The formulatio of this fuctio Val(A i ) is strogly depedet upo the type of iformatio assumed about the ucertai variable V. I the case whe we assume probabilistic ucertaity the geerally accepted approach is to use the expected value. We ote however this ca be called ito questio as we did i [7]. x x x 1 j A 1 A C i ij A m Figure #1. Decisio Matrix Decisio Attitude i Possibilistic Decisio Makig We shall first cosider the classic case of possibilistic ucertaity ad discuss the role of decisio attitude. I this situatio i which all we kow about the ucertai variable is that it must take its value i the set X = {x 1, x,... x }, called the feasible set. No iformatio is assumed distiguishig the prospects of the elemets i X. From a decisio makig poit of view alterative A i iduces a collectio of possible [C i1, C i,..., C i ]. A umber of differet approaches have bee suggested for evaluatig alteratives i this eviromet. Amog these are the followig:

2 1). Pessimistic: Val(Ai) = Mi j [Cij] 1). If W = W where w * j = 0 for j = 1 to - 1 ad ). Optimistic: Val(Ai) = Max j [C ij ] w = 1, we get Val(A i ) = Mi j [C ij ], the pessimistic 3). Average/eutral: Val(A i ) = 1  decisio maker. C ij ) If W = W *, where w 1 = 1 ad w = 0 for j = to j 4). Arrow-Hurwicz Criteria:, we get Val(A i ) = Max j [C ij ] the optimistic decisio Val(A i ) = a Max i [C ij ] + (1 - a) Mi i [C ij ] (for a[0, 1]) maker.. Each oe of these ca be see as reflectig a particular 3).If W = W A, where w 1 attitude o the part of the decisio maker. For example, j = for all j we get Val(Ai) = 1  i the first oe the decisio maker is essetially displayig a pessimistic attitude, oe i which he is C i. the eutral decisio maker assumig that the worst of the possible outcomes will occur. Thus we see that a sigificat aspect of these 4) If W = W H, where w 1 = a, w = 1 - a ad w j = methods is the cetrality of the choice of decisio 0 for j = to - 1 we get Val(Ai) = amax j [C ij ] + attitude i the process of valuatig alteratives. Also, (1 - a) Mi j [Cij], the Arrow-Hurwicz criteria. quite clear i this situatio is the subjectiveess of the I [8] we referred to W as the attitudial vector ad choice. itroduced a measure of optimism associated with a I [8] we provided for a uificatio ad give vector: a(w) = 1  geeralizatio of these approaches to decisio makig. w j ( - j). We ote - 1 The approach is based upo the OWA operator [9, 10]. We first describe the OWA operator. that a (W * ) = 0, a (W*) = 1, a(w A ) = 0.5 ad A Ordered Weighted Averagig (OWA) operator a(w H ) = a of degree is a mappig F I additio to uifyig the existig decisio W : R Æ R which has a makig techiques, the itroductio of the OWA associated vector weightig vector W, whose formalism provides for a geeralizatio of the potetial compoets satisfy meas we have for evaluatig alteratives. The W-1:. w j Œ [0,1] for j = 1 to geeralizatio itroduced here is oe i which by selectig a W we get a valuatio fuctio. This of W-:.  w j = 1 course provides us with a ifiite family of possible valuatio fuctios. It would be useful here to address ad where F w (a 1, a,..., a ) = the issue of providig some directio for the process of  w j b j with b j selectig a attitudial vector W. The more deep our uderstadig of the valuatio methodology, the more beig the j th largest of the a i. itelligetly we ca select the attitudial vector W. Essetially, the OWA operator ivolves a I [11] Yager suggested a iterestig iterpretatio reorderig of the argumets, the a liear aggregatio. of the valuatio process. Notig that the elemets i the attitudial vector W, as required by W-1 ad W-, Thus if B is the vector of reordered argumets the we satisfy the properties of a probability distributio, they ca express this as lie i the uit iterval ad sum to oe, he suggested F w ( a 1, a,..., a ) = W T B, that the compoets i W ca be iterpreted as a kid of it is the ier product of the weightig vector ad the subjective probabilities. I particular, he suggested that vector of ordered argumets. w j ca be iterpreted as the decisio maker's subjective I [8] we suggested the use of the OWA operator as probability that the j th best outcome will occur. With a structure for providig a uificatio ad geeralizatio this iterpretatio, the pessimist is sayig, sice i this of the decisio makig formalisms used i decisio case w = 1, that the probability that the worst thig makig uder possibilistic ucertaity. Assume A i is a will happe is oe while the optimist is sayig that the decisio alterative with payoff bag [Ci1, C probability that the best thig will happe is oe. i,..., Uder this iterpretatio of the w j 's, as these type of Ci] the we suggested usig Val(Ai) = F W (Ci1, C i,..., Ci) as the valuatio of the alterative. By probabilities ad with b j beig the j th best payoff, we appropriately selectig W we ca obtai ay of the previously itroduced evaluatio fuctios.

3 are the ordered payoffs for alteratives A see the the evaluatio Val(A i ) =  1 ad A b j w j is a kid of respectively. Let us look at differet kids of decisio makers. expected value. (1) W T = 1 1 I order to use this OWA approach for valuatig 0 0, this is a strogly alteratives, we eed select a attitudial vector W. A optimistic decisio maker. I this case umber of approaches have be suggested for the determiatio of the attitudial vector W. Here we Val(A 1 ) = 1 ( ) = 80 describe oe approach. I this approach, all that is Val(A ) = 1 ( ) = 75 required of the decisio maker is that they provide a value a Œ [0, 1] idicatig their desired degree of Here A 1 is the preferred alterative.. optimism. Usig this value the weights for the () If W T = , strogly pessimistic appropriate attitudial vector ca be obtaied by solvig the followig mathematical programmig problem. decisio maker. I this case Val(A 1 ) = 1 ( + 30) = 40 Maximize:  w j l(w j ) Val(A ) = 1 ( ) = 55 Subject to: Here A is preferred. 1.w Œ [0, 1] j (3) For the eutral decisio maker, W.  w j = 1 = we get Val(A 1 ) - 1 ( ) = w j ( - j) = a Val(A - 1 ) = 1 ( ) = 65 4 I the above we see that costraits 1 ad are the Here A is agai preferred. basic costraits for the OWA vector. Costrait 3 (4) If we cosider a decisio maker with a = 0.65 assures us that the degree of optimism of the determied ad use the weights obtaied by solvig the precedig vector satisfies our desired coditio of a. The mathematical programmig problem, we have objective fuctio ca be see as oe of maximizig the W T = etropy of the determied weights. This approach ca I this case be see as a geeralizatio of the Arrow-Hurwicz criteria Val(A 1 ) = (4)100 + (0.8)(60) + (0.19)) [1]. I both approaches the decisio maker supplies a + (13) = 70. degree of optimism. I the Arrow-Hurwicz criteria, we Val(A ) = (0.4)80 + (8)(70) + (0.19))60 oly allocate weight to the maximal ad miimal + (13) = 69.5 payoffs associated with a alterative. I this approach the weight is distributed to all the payoffs associated Decisio Makig with D-S Belief with the alterative beig evaluated. Structures The followig example illustrates the use of the We have cosidered decisio makig i situatios OWA operator. i which the iformatio about the ucertai variable V x 1 x x 3 x 4 is possibilistic. The Dempster-Shafer belief structures provides aother ucertai kowledge represetatio. Example: A We first provide a formal characterizatio of the A Here Dempster Shafer belief structure. A Dempster-Shafer belief structure defied o a set X has a associated collectio of subsets of X, B j for B 1 = j = 1 to, called focal elemets ad a mappig m associatig values with subsets of X, ad B = m: 60 Æ [0, 1] such that: 30

4 1). at some future date. Assume iterest rates are  m(b j ) = 1 determied by the Federal Reserve Board. Assume that the board is cosiderig three policies: make iterest ). m(a) = 0 for A ot a focal elemet. rates low, make iterest rates high ad ot iterfere with Two measures that are defied o these belief iterest rates. We believe that there is a % chace structures, which we shall fid useful, are the measure that the first policy will be followed, 0% chace of the of plausibility ad belief. The plausibility measure Pl secod ad a 30% chace of the third. We ca represet is a mappig Pl: X Æ [0,1] defied as the iformatio about the variable V, future iterest rates, as a belief structure i which B 1 is the subset of Pl(A) =  all j s.t. A «B j π m(b j ). The measure of belief, Bel, is a mappig o subsets of Xdefied by Bel(A) =  m(b j ) all j s.t. A Õ B j It ca be show that Bel(A) = 1 - Pl(A). A importat special case of belief structures are X, the set of iterest rates, cosistig of low iterest rates, B is the subset of X cosistig of high iterest rates ad B 3 = X, ay iterest rate. Give a D-S belief a issue of cocer is the determiatio of the probability that the value of V lies i some subset A of X; Prob(A). Geerally, because of the type of ucertaity modeled by the D-S this value ca't be precisely obtaied. The best we ca do is to fid upper ad lower bouds, Prob + (A) ad Prob - (A) called cosoat belief structure i this case the focal such that elemets are ested, that is they ca be idexed so that Prob - (A) Prob(A) Prob + (A) B 1 à B... à B. I this case the plausibility It ca be show that these bouds are related to the measure has the special property that measures of of plausibility ad belief itroduced earlier, Pl(E» F) = Max[Pl(E), Pl(F)] i particular Prob + (A) = Pl(A) ad Prob - (B) = Bel(A). A various sematics ca be associated with the D-S Usig this D-S represetatio various types of the belief structure. We shall fid the followig sematics, kowledge ca be represeted for example the oe very much i the spirit of radom sets, useful for kowledge that that Prob(A) a is represetable by a our curret iterest, decisio makig uder ucertaity. belief structure i which B 1 = A ad B = X ad m(b 1 ) Let V be a variable that takes its value i the set X. = a ad m(b ) = 1 - a. The kowledge that Prob(A) Assume our kowledge about the value of this variable is that it is determied at least i part by performace of = a is represetable i this structure by a belief a biased radom experimet. I this radom experimet structure i which B 1 = A ad B = A ad where the outcomes, rather beig elemets of X are subsets of m(b 1 ) = a ad m(b ) = 1 - a. the space X. Furthermore, it is assumed that the actual We shall ow tur to the issue of decisio makig value of V must be a elemet of the set that is i situatios i which our kowledge about the determied by the radom experimet. We let p j be the ucertai variable is represeted by a D-S belief probability that F j is the outcome subset of the structure. experimet. If we assig F Assume we have a decisio problem where our j = B j ad let m(b j ) = p j the we ca represet this as a D-S belief structure. kowledge about the ucertai variable V is represeted Probabilistic ucertaity is a special case of this by a belief structure with focal elemets F 1,..., F q structure i which the focal elemets are sigletos, with weightig fuctio m. I [8] we suggested a B geeral approach to this decisio problem based upo a j = {x j } ad m(b j ) = p j, the probability of x j..the determiatio of a valuatio Val(A i ) for each possibilistic ucertaity previously discussed is a special case of a belief structure, it is oe i which alterative. The procedure used for the valuatio of a alterative is give i the followig. I order to use B 1 = X ad m(b 1 ) = 1. this approach the decisio maker must provide a value The ext example is a illustratio of a case i a Œ [0, 1] correspodig to the degree of optimism which we have kowledge of a ucertai variable that they wat to use i the decisio problem. This degree is ot represetable by the possibilistic or probability of optimist will be used to geerate attitudial model but is represetable by the D-S belief structure. weightig vectors usig algorithm of the earlier sectio. Let V be a variable correspodig to iterest rates I the followig we shall use W a,r to idicate the

5 attitudial vector of dimesio r geerated from this Val(A 1 /3) = (70)(0.75) + ()(0.5) = 65 algorithm based upo a degree of optimism. The Fially, procedure evaluatig alterative A i is as follows: 3 1). For each focal elemet F j calculate Val(A i /j ) Val(A 1 ) =  Val(A 1 /j ) p j where Val(A i /j ) = (W a, ) T B j j. Here j is the = (90)(0.5) + (65.1)(0.3) + (65)(0.) = cardiality of the focal set F j, B j is the vector If we used a = 0.5 istead of 0.75 the for cosistig of the ordered payoffs that are associated with = : w 1 = w = 1 the elemets i F j ad the alterative A i. (Val(A i /j ) is a OWA aggregatio with attitudial vector W = 3: w 1 = w = w 3 = 1 a,j ad 3 I this case argumets C ik for all x k Œ F j ). Val(A 1 /1 ) = 1 ( ) = 80 ) Calculate Val(A i ) as the expected value of the Val(A i /j) that is Val(A i ) =  Val(A i/ j ) p j Val(A 1 / ) = 1 ( ) = 60 3 Val(A 1 /3 ) = 1 (70 + ) = 60 Here Val(A 1 ) = (80)(0.5) + (60)(0.3) + (60)(0.) = 70. here of course p j = m(b j ). Notice the valuatio for a = 0.5 is less the the case i We see that i step oe we used the possibilistic which we used a = 0.75 this is because it is more techique ad i step two we used the expected value pessimistic. characteristic of probabilistic eviromet. We ca calculate Val(A ) i a similar maer, The followig example illustrates the procedure however we shall ot do this. just described. Example: Agai we assume the followig decisio Graded Possibilistic Decisio Makig matrix: Possibilistic ucertaity is ofte geerated from x 1 x x 3 x 4 liguistic iformatio. A example of this is the A statemet "iterest rates will be low." Here we obtai A as the set of possible iterest rates all those cosidered We assume kowledge of the ucertai variable V as low. With the itroductio of Zadeh's work o fuzzy is expressed by the followig belief structure with three sets [13] we have become much more aware of the fact focal elemets: F that cocepts such as "low iterest rates" rather tha 1 = {x 1, x }, F = {x, x 3, x 4 }, beig crisp ad well bouded are imprecise ad fuzzy. F 3 = {x 3, x 4 } where m(f 1 ) = 0.5, m(f ) = 0.3 ad I particular, the set of low iterest rates rather tha m(f 3 ) = 0.. beig a crisp set is i reality a fuzzy subset. This Here we shall assume our degree of optimism a = observatio requires us to have to come to grips with Usig our algorithm we obtai the followig graded possibilistic iformatio. I particular, attitudial vectors for = ad 3. iformatio about a variable V based o fuzzy subsets = : w iduces a possibility distributio P [1] o the domai 1 = 0.75 w = 0.5 of V where for each x i the domai P(x) Œ [0, 1] = 3 w 1 = 0.6 w = 0.7 w 3 = 0.11 idicates the possibility that x is the value of the For F 1, F, ad F 3 the collectio of associated payoffs variable V. We shall at this poit ot digress ito the are respectively [100, 60], [60, 70, ] ad [70, ]. theory of possibility, however we ote that [3, 14] These result i followig ordered payoff vectors:. provides a comprehesive itroductio to the field. Here we shall oly cosider the problem of decisio B 1 = B = 60 B 3 = 70 makig with graded possibilities. Thus we shall assume for each x Œ X there exists a value P(x) Œ [0, 60 1] idicatig the possibility that x is the value of V. Usig this we get As with all cases of possibilistic decisio makig Val(A we must provide some idicatio of the decisio makers 1 /1) = (100)(0.75) + (60)(0.5) = 90 attitude, here we shall assume a decisio maker with Val(A 1 /) = (70)(0.6) + (60)(0.7) + (0.11) =65.1 degree of optimism a. For otatioal simplicity we

6 shall use OWA a (G) to idicate the aggregatio of the 1 elemets i the set G usig the attitudial vector Val(A 1 ) = A 1/u du = (70)(0.3) + (73.3)(0.) + geerated by a degree of optimism a ad dimesio equal to the cardiality of G. 0 (85)(0.3) + (70)(0.) = I calculatig Val(A i ) i this case of graded A atural extesio here is to cosider the cases i possibilistic ucertaity rather the treatig all which we have graded possibility i Dempster-Shafer outcomes similarly we eed give more weight to those belief structures. I this case the focal elemets rather outcomes havig larger grade of possibility. The the beig crisp subsets of the domai would be fuzzy followig is a procedure for evaluatig Val(A i ) i this subsets of the domai. Here each focal elemet F j eviromet of graded possibility would iduce a graded possibility distributio P j o the 0. Determie the largest degree of possibility for domai of the ucertai variable i which P j (x) = F j (x), ay elemet i the domai X of the ucertai variable the membership grade of x i the fuzzy focal elemet. ad deote this as U max. The valuatio process i this case would require that i 1. For each u Π[0, U max ] deote F u as the calculatig Val(A i/j ), the valuatio of the alterative subset of X for which P(x) u. F u are sometimes with respect to the jth focal elemet, we use the called level sets. procedure described i this sectio for graded ). For each F u obtai the bag of associated possibilistic decisio makig. payoffs for the alterative beig evaluated, B u. Thus if Coclusio x k ΠF U the C ik is i the bag B u. We believe that possibility theory ad D-S theory 3). Calculate Val(A i/u ) = OWA (B a u ) provide useful frameworks for the represetatio of ucertaity i maagemet chace discovery. Here we U max 4) Val(A i ) = 1 have focused o the issue of decisio makig uder OWAa du these types of ucertaity. We particularly oted the Umax 0 importace of the role of decisio attitude. The followig example illustrates this approach. Example: Assume the payoffs for alterative A i is A x 1 x x 3 x 4 ad we have the followig iformatio o graded possibilities P(x 1 ) = 0.8, P(x ) = 0.3, P(x 3 ) = 1, P(x 4 ) = 0.5 I this case Referece [1]. Zadeh, L. A., "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets ad Systems 1, 3-8, []. Dubois, D. ad Prade, H., Possibility Theory : A Approach to Computerized Processig of Ucertaity, Pleum Press: New York, [3]. De Cooma, G., Rua, D. ad Kerre, E. E., Foudatios ad Applicatios of Possibility Theory, F u = {x 1, x, x 3, x 4 } u 0.3 World Scietific: Sigapore, F u = {x 1, x 3, x 4 } 0.3< u 0.5 [4]. Yager, R. R., "O the istatiatio of possibility distributios," Fuzzy Sets ad Systems 18, 61-65, F u = {x 1, x 3 } 0.5 < u F u = {x 3 } 0.8 < u 1 [5]. Shafer, G., A Mathematical Theory of Evidece, ad therefore, Priceto Uiversity Press: Priceto, N.J., B u = {100, 60, 70, } u 0.3 [6]. Yager, R. R., Kacprzyk, J. ad Fedrizzi, M., B u = {100, 70, } 0.3 < u 0.5 Advaces i the Dempster-Shafer Theory of Evidece, B u = {100, 70} 0.5 < u 0.8 Joh Wiley & Sos: New York, B u = {70} 0.8 < u 1 [7]. Yager, R. R., "Fuzzy modelig for itelliget For simplicity we shall assume a eutral decisio decisio makig uder ucertaity," IEEE Trasactios maker, a = 0.5. Thus at each level we take the average. o Systems, Ma ad Cyberetics Part B: Cyberetics Val(A 1/u ) = 70 u , 60-70, 000. Val(A [8]. Yager, R. R., "Decisio makig uder Dempster- 1/u ) = < u 0.5 Val(A Shafer ucertaities," Iteratioal Joural of Geeral 1/u ) = < u 0.8 Systems 0, 33-45, 199. Val(A 1/u ) = < u 1

7 [9]. Yager, R. R., "O ordered weighted averagig aggregatio operators i multi-criteria decisio makig," IEEE Trasactios o Systems, Ma ad Cyberetics 18, , [10]. Yager, R. R. ad Kacprzyk, J., The Ordered Weighted Averagig Operators: Theory ad Applicatios, Kluwer: Norwell, MA, [11]. Yager, R. R., "O the iclusio of importaces i OWA aggregatios," i The Ordered Weighted Averagig Operators: Theory ad Applicatios, edited by Yager, R. R. ad Kacprzyk, J., Kluwer Academic Publishers: Norwell, MA, 41-59, [1]. Arrow, K. J. ad Hurwicz, L., "A optimality criterio for decisio makig uder igorace," i Ucertaity ad Expectatios i Ecoomics, edited by Carter, C. F. ad Ford, J. L., Kelley: New Jersey, 197. [13]. Zadeh, L. A., "Fuzzy sets," Iformatio ad Cotrol 8, , [14]. Klir, G. J. ad Yua, B., Fuzzy Sets ad Fuzzy Logic: Theory ad Applicatios, Pretice Hall: Upper Saddle River, NJ, 1995.