Causality Consistency Problem for Two Different Possibilistic Causal Models

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1 nernaonal ymoum on Medcal nformac and Fuzzy Technoloy (MF Hano (Au. 999 Caualy Conency Problem for Two Dfferen Poblc Caual Model Koch YAMADA Dearmen of Plannn and Manaemen cence Naaoka Unvery of Technoloy 603- Kam-omoka Naaoka Naa JAPAN E-mal: ABTRACT Condonal Caual Probably (CCPR and Condonal Caual Pobly (CCPO have been rooed o exre exac uncerane of cauale and ome reaonn mehod baed on hem have been uded o calculae robable or oble of unknown even under he condon ha ome even are known. CCPR/CCPO a condonal robably/obly of a cauaon even condoned by caue. Cauaon even an Òeven ha a caue acually caue an effec.ó The aer rooe o clafy caual model baed on cauaon even no wo ye -- ymmercally valued and aymmercally valued caual model -- deendn on he roere of varable whch ake an even a her value. alo how he relaon beween CCPO and convenonal condonal oble n hee wo ye of caual model. Then defne and dcue a Caualy Conency Problem whch a roblem o calculae he obly of combned value of arbrarly choen unknown varable when value of ome oher varable are known.. NTRODUCTON Condonal Caual Probably (CCPR and Condonal Caual Pobly (CCPO have been rooed o exre exac uncerane of cauale and ome reaonn mehod baed on hem have been uded o calculae uncerane of unknown even when ome oher even are known [-5]. CCPR defned a condonal robably of a cauaon even condoned by caue. Cauaon even an "even ha a caue are and acually caue an effec [-3]." CCPO a oblc veron of he dea [45] whch emloy Pobly heory [67] o exre uncerane nead of Probably heory. The aer fr rooe o clafy caual model baed on cauaon even no wo ye -- ymmercally valued and aymmercally valued caual model -- deendn on he roere of varable whch ake an even a her value. alo how ha he relaon beween CCPO and convenonal condonal oble are dfferen beween hee wo ye of model. Then defne a Caualy Conency Problem and dcue oluon and her roere. The defned roblem one o calculae he obly of combned value of arbrarly choen unknown varable when value of ome oher varable are known. The roblem an exended one ncludn nvere roblem of caualy [4] and caualy analy roblem [5] uded n he revou reearch. Wha hould be noced n he dcuon for he roere of oluon ha we do no have o be cauou wh he dfference of wo caual model. Th becaue he dcuon oble un convenonal condonal oble derved from CCPO exce a roof of a rooon. A he end of he aer a numercal examle of a ymmercally valued model hown. The examle verfe ome of he roere of oluon dcued n he aer. 2. TWO DFFERENT CAUAL MODEL Fr le u defne caue effec and cauaon even whch were nroduced n [-3]. [Defnon 2.] Le x (É and y (=ÉJ be varable akn a value n a e of even U ={u Éu H and V ={v Év K reecvely. The value of y deenden on hoe of ome x. Thu u h (h=éh and v k (k=ék are called caue and effec even reecvely. x ( y a caue (an effec varable. Then cauaon even one ha Òu h are and u h acually caue v k Ó and exreed n v k :u h 82

2 afyn he follown equaon: v : u «( v : u u «( v : u v «( v : u u v k h k h h k h k k h h k ( v : u u «u (2 k h h h v : u v «v (3 k h k k where «and mean equvalence and local roduc. u h denoe he abence of u h. When v k :u h rue ad ha u h uor v k. Then we defne he ymmercally valued caual model menoned n NTRODUCTON. [Defnon 2.2] When x Î{ u... uh and y Î{ v... vk afy he nex equaon x and y comoe a ymmercally valued caual (VC model. vk «( vk : uh. (4 h The above defnon how ha an effec even are f and only f caued by one or more of caue even. Th called Mandaory Cauaon Aumon n [5] houh he word wa ued a fr n aymmercally valued caual model a decrbed below [3]. Now we conder a cae where H=K=2 ha U ={u u 2 and V ={v v 2. f we aume u = u and v = v we can rewre U = { u u and V = { v v becaue U and V are excluve and exhauve e. Then he aymmercally valued caual model defned below. [Defnon 2.3] When x Î{ u uand y Î{ v v afy he follown equaon x and y comoe an aymmercally valued caual (AVC model. v «( v : u. (5 v : u «v : u «v : u «F (6 where F mean fale. n he above eq. (6 mean ha u canno uor any v and ha v canno be uored by any caue even. Therefore obvou ha eq. (4 doe no hold for v = v 2. nead we e he nex from eq. (5. v «( v : u (7 For an AVC model Mandaory Cauaon Aumon mean ha eq. (5 hold [3]. The dfference beween a VC and an AVC model no merely he number of oble value aken by varable. n a VC model every effec even v k become rue only when uored by one or more caue even and here only a k where v k rue for any. An AVC model on he oher hand doe no requre any uor for v. v become rue (ha v become fale when v no uored by any caue even. For examle le x x 2 and y mean Weaher eabed Color-of-ea-urface reecvely. Ther oble value mh be {fne cloudy rany {and rock coralreef and {blue reen ray. f he color of ea urface deenden on he weaher and/or he eabed x x 2 and y comoe a VC model. A an examle of an AVC model uoe ha x mean deae and ha y are her ymom. Thee varable ake rue ( v or fale ( v a her value for he deae and he ymom. n h cae x and y comoe an AVC model becaue ymom do no aear whou any deae. 3. CONDTONAL CAUAL POBLTY n Defnon 2. menoned ha he value of an effec varable deenden on hoe of ome caue varable. The deendence however omeme unceran n he real world. n order o coe wh he uncerany we nroduce Pobly heory [67] no he caual model. 3. General Proere of Poble Le w and w 2 be varable akn a value n a e W and W 2 reecvely. Marnal obly drbuon on W and W 2 are exreed n (w and (w 2. Then he follown equaon hold n eneral [67]. 83

3 ( w ( w2 ³ ( w w2 (8 ( w w = ( w ( w w (9 2 2 where ( mean local roduc (local um when ued for even whle mean mn (max for obly drbuon. ( w w2 equvalen o on obly drbuon ( w w2. Poblc ndeendence and non-neracon are defned a follow: [Defnon 3.] w 2 oblcally ndeenden of w f and only f he nex equaon hold. ( w2 w = ( w2. (0 [Defnon 3.2] w and w 2 are non-neracve f and only f he nex equaon hold. ( w w2 = ( w ( w2. ( From eq. (9 and (0 w and w 2 are non-neracve f w 2 (w oblcally ndeenden of w (w Condonal Caual Pobly for a VC model Th econ defne Condonal Caual Pobly (CCPO for a VC model and examne he relaon beween convenonal condonal oble and CCPO. Noe ha x and y comoe a VC model hrouhou he econ. [Defnon 3.3] Le (x and (y be marnal obly drbuon on U and V reecvely. (y :x a obly drbuon of y :x on U V. Then ( y : x x called Condonal Caual Pobly drbuon for a VC model. [Defnon 3.4] n cae of a VC model x and y have caualy f and only f he nex equaon hold. $ ( hk ; ( vk : uh h > 0 uh ÎU vk ÎV. (2 [Defnon 3.5] n cae of a VC model a conuncon of cauaon and caue even a conex of y :x f nclude neher caue even of x nor cauaon even of y :x. [Defnon 3.6] Le X be a conex of y :x n a VC model. y :x oblcally cauaon ndeenden f and only f he nex equaon hold. ( y : x x X = ( y : x x. (3 Then we aume he follown: (a n cae of a VC model x (É oblcally ndeenden of each oher. (b n cae of a VC model y :x (É =ÉJ oblcally cauaon ndeenden. From he above aumon we can derve he follown rooon [5]. [Prooon 3.] n cae of a VC model y :x and x Õ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.2] n cae of a VC model y :x and y Õ :x Õ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.3] n cae of a VC model he nex equaon hold. ( y : x y : x Y = ( y : x Y ( y : x Y (4 where Y a conuncon of caue even and a neceary condon. ¹ no From he above he follown equaon are derved [5]. ( ( y = ( y : uh h ( uh (5 uh ÎU... ( y x... x = ( y : x x ( ( y : uh h ( uh (6 uh ÎU + ( y... yq x... x = ( y x... x (7 = q 84

4 where and q J. oblcally cauaon ndeenden f and only f he nex equaon hold. 3.3 Condonal Caual Pobly for an AVC model The econ defne CCPO for an AVC model and examne he relaon beween condonal oble and CCPO. Throuhou h econ x and y comoe an AVC model. [Defnon 3.7] Le obly drbuon on U = { u u V = { v v and Z = { v : u v : u be (x (y and (z reecvely. Then ( z x called Condonal Caual Pobly drbuon for an AVC model. Pleae noe ha CCPO drbuon for an AVC model defned on U Z no on U V lke a VC model. From eq. (6 n Defnon 2.3 we can e he follown: ( v : u = ( v : u = ( v : u = 00.. (8 When u «F ( u «T he follown are derved from eq. ( and (2. v : u «v : uu T «v : uu u «F. (9 v : u «v : u T «v : u u «u «T. (20 Therefore we e he follown CCPO. ( v : u = 00.. (2 ( v : u = 0.. (22 [Defnon 3.8] n cae of an AVC model x and y have caualy f and only f he nex equaon hold. ( v : u > 0. (23 ( z x X = ( z x. (24 Then we aume he follown: (a n cae of an AVC model x (É oblcally ndeenden of each oher. (b n cae of an AVC model z (É =ÉJ oblcally cauaon ndeenden. From he above aumon we can derve he follown rooon [4]. [Prooon 3.4] n cae of an AVC model z and x Õ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.5] n cae of an AVC model z and z ÕÕ ( ¹ are oblcally ndeenden of each oher. [Prooon 3.6] n cae of an AVC model he nex equaon hold. ( z z Y = ( z Y ( z Y (25 where Y a conuncon of caue even and a neceary condon. ¹ no From he above he follown equaon are derved [4]. ( = ì ( v : u ( u f y v ì ü ( y = íí ( ( v : u x ( x ý îxî{ u u þ f y = v î (26 [Defnon 3.9] n cae of an AVC model a conuncon of cauaon even caue even and/or her neaon a conex of z f nclude none of u u v : u and v : u. [Defnon 3.0] Le X be a conex of z n an AVC model. z 85

5 ( y x... x ì ( v : u x ( ( v : u ( u f y = v + (27 = í ( v : u x ì ü í ( ( v : u x ( x ý + îxî{ u u þ î f y = v ( y... yq x... x = ( y x... x (28 = q where and q J. Po( P QT P Q º ( x y... u v... vq = + = q+ T x ÎP Í X - P T J y ÎQT ÍY -Q (29 where u... u v... v are he oberved value. q The above CCP reduced o an nvere roblem of caualy uded n [4] when P =Æ P = X and Q T =Æ. Caualy Analy Problem dcued n [5] a ecal cae where P =Æ and Q T = 0 or P =Æ and Q T =Æ. n h ene CCP an exended roblem o analyze a wo-layered herarchcal caual model. Un he roery of eq. (0 he nex equaon derved. 4. CAUALTY CONTENCY PROBLEM Th chaer defne a Caualy Conency Problem whch alcable boh o a VC and an AVC model and dcue he oluon. The dcuon can be conduced wh condonal oble derved from CCPO n he way hown n he revou chaer. n oher word we do no have o be cauou wh he dfference beween he wo model hrouhou he dcuon exce he roof of Lemma Problem Defnon and oluon Caualy Conency Problem defned a follow. [Defnon 4.] Le X be a e of caue varable x (É and Y be a e of effec varable y (=ÉJ. Marnal obly drbuon of he caue varable and CCPO drbuon are ven a a ror knowlede (Noe ha he defnon of CCPO are dfferen beween a VC and an AVC model. Then uoe ha value of x Î P ={ x... x and y Î Q ={ y... y q q J are oberved and ha hoe of he oher varable are unknown. Caualy Conency Problem (CCP one o oban a obly of combned value of ome unknown varable n P = { x+... x Í X -P and QT = { yq+... yt ÍY -Q under he ven condon. ( x y... u v... vq = + = q+ T ( v... q... ( v u u u... u (... q = v v y... u u = q+ T = + ( u... u u. = + (30 Le he 2nd and 3rd erm n he lef de of he equaon be E(PQ and he rh de be H(PQP Q T. Then hee are exreed a follow reardle of whch caual model baed on a VC or an AVC model houh he way o calculae he condonal oble are dfferen a dcued before. EPQ ( = ( v... u ( u. (3 = q HPQP ( QT = ( v... u x = q = + ( y... u x = q+ T = + ( u ( x. = + (32 E(PQ he obly ha u... u v... v are oberved. H(PQP Q T ve he obly ha he oberved value and value ecfed by x Î P and y Î QT haen mulaneouly. Un E(PQ and H(PQP Q T eq. (30 rewren a follow: q 86

6 PoP ( QT PQ EPQ ( = H( PQP QT. (33 Ulzn he dea of he lea ecfc oluon [8] whch chooe he oluon wh he reae obly deree n areemen wh he conran by eq. (33 he oluon of Caualy Conency Problem obaned n he nex equaon. Po( P QT P Q ìhpqp ( QT f EPQ ( > HPQP ( QT (34 = í î f EPQ ( = HPQP ( QT. 4.2 ome Proere of oluon ome roere of he oluon ven by eq. (34 are dcued. Thee roere hold n boh cae of a VC and an AVC model. [Prooon 4.] Po({ x Æ P Q Po( Æ{ y P Q xîp yîqt ³ Po( P QT P Q (35 Proof: The defnon ven by eq. (29 how ha eq. (35 hold f ( w w3 ( w2 w3 ³ ( w w2 w3. Aume ha ( w w3 ( w2 w3 < ( w w2 w3. Thu ( w w3 ( w2 w3 ( w3 < ( ww2 w3 ( w3 hold for w 3 afyn ( w3 > ( w w3 ( w3 > ( w2 w3 and ( w3 ³ ( w w2 w3. Thouh h lead o ( ww3 ( w2 w3 < ( ww2 w3 conradc eq. (8. Th becaue he aumon fale. (EOP [Prooon 4.2] Po( Æ { y P Q = Po( Æ QT P Q (36 yîqt Proof: Le QT y y Po( Æ{ y y P Q = {. Then he nex hold. ìhpq ( Æ { y y f EPQ ( > HPQ ( Æ{ y y = í î f EPQ ( = HPQ ( Æ{ y y. From eq. (32 we e H( PQ Æ { y y = H( PQ Æ{ y H( PQ Æ { y. Therefore he nex equaon hold. Po( Æ { y y P Q = Po( Æ{ y P Q Po( Æ { y P Q. Th clearly lead o eq. (36. (EOP [Lemma 4.] Le P X P Í - P X P 2 Í - Q X Q T Í - and Q X Q T 2 Í -. Then he follown hold. HPQP ( QT = HPQP ( QT 2 2 «Po( P QT P Q = Po( P QT P Q 2 2 HPQP ( QT > HPQP ( QT 2 2 «Po( P QT P Q > Po( P QT P Q 2 2 Proof: Thee are ealy roved from eq. (34. (EOP [Lemma 4.2] The nex equaon hold for x x ÎX - P. HPQ ( { x HPQ ( { x x. (37 (38 Æ ³ Æ (39 Proof: (a Cae of a VC model Le F({x x be defned a F({ x x = ( y : u { h h h uh ÎU ¹ = + ( y : u ( u. Thu he follown are derved from eq. (6. ( y... u x x ( x ( x = [ F({ x x ( y : x x ( y : x x ] ( x ( x éf({ x x { ( y : x x ( x ù = { ( y : x x ( x ( x ( x. ( y... u x ( x éf({ x x { ( y : x x ( x ù = ì ü ( x. í ( ( y : uh h ( uh ý î uh ÎU þ From he above he nex equaon obaned. ( y... u x ( x ³ ( y... u x x ( x ( x. Condern eq. (32 obvou ha eq. (39 hold n cae of a VC model. (b Cae of an AVC model (b- When y = v le G ({x x be defned a G ({ x x = ( v : u { ¹ = + ( v : u ( u. Thu he follown are derved from eq. (27. 87

7 ( v... u x x ( x ( x = G({ x x ( v : u x ( v : u x ( x ( x ég ({ x x { ( v : u x ( x ù = { ( v : u x ( x ( x ( x. ( v... u x ( x [ ] [ { ] = G ({ x x ( v : u x ( v : u ( u ( x ég ({ x x { ( v : u x ( x ù = ( x. { ( v : u ( u Condern x = u or u and ( v : u = 00. he nex equaon hold. ( v... u x ( x ³ ( v... u x x ( x ( x. (b-2 When y = v le G 2 ({x x be defned a G ({ x x = ( v : u 2 ¹ + ì í î xî{ u u ü ( v : u x ( x ý. þ ( Thu he follown are derved from eq. (27. ( v... u x x ( x ( x { = G ({ x x ( v : u x ( x 2 { ( v : u x ( x. ( v... u x ( x = G ({ x x ( v : u x ( x 2 { ì ü í ( ( v : u x ( x ý. îxî{ u u þ Therefore he nex equaon hold. ( v... u x ( x ³ ( v... u x x ( x ( x. From he reul of (b- and (b-2 eq. (39 alo hold n cae of an AVC model. (EOP [Prooon 4.3] Po({ x Æ P Q ³ Po( P Æ P Q. (40 xîp Proof: obvou from lemma 4. and 4.2. (EOP [Prooon 4.4] Po( P Æ P Q Po( Æ Q P Q ³ Po( P Q P Q. (4 T T Proof: Th roved f he follown wo nequale hold. HPQP ( Æ ³ HPQP ( Q. T HPQ ( Æ QT ³ HPQP ( QT. The former obvou from eq. (32. The laer roved n a mlar way o lemma 4.. (EOP 5. NUMERCAL EXAMPLE Le u verfy ome roere dcued n econ 4.2 un a numercal examle of a VC model hown n [5]. Le X = { x x2 x3 U = { a b c( = 23 Y = { y y2 y3 and V = { d e f( = 23. ( x and ( y : u are ven a follow: P ( a ( b ( c = [ ]= [ ] = [ ( a ( b ( c ]= [... ] = [ ( a ( b ( c ]= [... ] P P é( ad ( ae ( a f ù é P = ( bd ( be ( b f = ( cd ( ce ( c f é3( ad 3( ae 3( a f ù é P 3 = 3( bd 3( be 3( b f = ( cd 3( ce 3( c f é2( ad 2( ae 2( a f ù é P 2 = 2( bd 2( be 2( b f = ( cd 2( ce 2( c f é22( ad 22( ae 22( a f ù é P 22 = 22( bd 22( be 22( b f = ( cd 22( ce 22( c f é32( ad 32( ae 32( a f ù é P 32 = 32( bd 32( be 32( b f = ( cd 32( ce 32( c f é33( ad 33( ae 33( a f ù é P 33 = 33( bd 33( be 33( b f = ( cd 33( ce 33( c f where ( a = ( x = a and ( ad= ( y = d: x = a x = a. Then uoe ha P = { x2 = b and Q = { y = e y3 = d are oberved. n h cae Condonal oble of combned value of ome unknown varable are obaned a follow un eq. (34. épo({ x = a Æ P Q ù é04 Po({ x Æ P Q = Po({ x = b Æ P Q = 0. Po({ x = c Æ P Q

8 épo({ x3 = a Æ P Q ù é0 Po({ x3 Æ P Q = Po({ x3 = b Æ P Q = 04. Po({ x3 = c Æ P Q 03. épo( Æ { y2 = d P Q ù é03 Po( Æ { y P Q = Po( Æ { y = e P Q = Po( Æ { y = f P Q Po({ x x3 Æ P Q épo({ x = a x3 = a Æ P Q ù é04 Po({ x = a x3 = b Æ P Q 04. Po({ x = a x3 = c Æ P Q 03. Po({ x = b x3 = a Æ P Q 0. = Po({ x = b x3 = b Æ P Q = 04. Po({ x = b x3 =c Æ P Q* 00. Po({ x = c x3 = a Æ P Q 02. Po({ x = c x3 = b Æ P Q 02. Po({ x = c x = c Æ P Q* Po({ x = b x3{ y2 P Q épo({ x = b x3 = a{ y2 = d P Q ù é03 Po({ x = b x3 = a{ y2 = e P Q 0. Po({ x = b x3 = a{ y2 = f P Q** 03. Po({ x = b x3 = b{ y2 = d P Q 03. = Po ({ x = b x3 = b{ y2 = e P Q = 04. Po({ x = b x3 = b{ y2 = f P Q** 03. Po({ x = b x3 = c{ y2 = d P Q 00. Po({ x = b x3 = c{ y2 = e P Q 00. Po({ x = b x3 = c{ y2 = f P Q 00. Po({ x x3 Æ P Q n he above how ha Prooon 4.3 afed. Cae wh aerk * are hoe where equaly doe no hold. hold n he oher cae. From Po({ x x3{ y2 P Q we can ee ha Prooon 4.4 afed. Cae wh ** are hoe where equaly doe no hold. We can alo verfy ha rooon 4. afed n he above examle. 6. CONCLUON The aer ndcaed ha caual model are clafed no wo dfferen ye - ymmercally and aymmercally value caual model - deendn on he roere of caue and effec varable when hey are modeled un cauaon even and condonal caual oble. alo howed ha he relaon beween condonal caual oble and convenonal condonal oble are dfferen beween hee wo model. Then Caualy Conency Problem defned and oluon were hown. ome roere of he oluon were alo dcued. A for he roere all of hem dcued n he aer hold boh n a VC and an AVC model becaue hey can be dcued n he level of condonal oble derved from condonal caual oble exce for a roof of a roery. Caualy Conency Problem an exended one n he ene ha nclude nvere roblem of caualy [4] a well a caualy analy roblem [5]. Therefore he rooed aroach could be alcable o many analycal roblem ha nclude uncerany due o he comlexy of he real world. REFERENCE [] Y. Pen J. A. Rea: A Probablc Caual Model for Danoc Problem olvn - Par : neran ymbolc Caual nference wh Numercal Probablc nference EEE Tran. y. Man and Cyber. Vol. MC-7 No (987 [2] Y. Pen J. A. Rea: A Probablc Caual Model for Danoc Problem olvn - Par : Danoc raey EEE Tran. y. Man and Cyber. Vol. MC-7 No (987 [3] Y. Pen J. A. Rea: Abducve nference Model for Danoc Problem-olvn rner-verla (990 [4] K. Yamada: Condonal Caual Pobly and Alcaon o nvere Problem of Cauale J. Jaan ocey of Fuzzy Theory and yem Vol. No (999 (n Jaanee [5] K. Yamada: Caual Reaonn Un Condonal Caual Poble o Exre Uncerane for Caualy Proc. FA99 (o aear [6] L. A. Zadeh: Fuzzy e a a ba for a heory of obly Fuzzy e and yem Vol (978 [7] E. Hdal: Condonal Poble ndeendence and Non-neracon Fuzzy e and yem Vol (978 [8] D. Dubo H. Prade: Poblc nference under Marx Form n H. Prade and C.V. Neoa (ed. Fuzzy Loc n Knowlede Enneern Verla T T Rhenland Kšln (986 89

(,,, ) (,,, ). In addition, there are three other consumers, -2, -1, and 0. Consumer -2 has the utility function

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