Evaluation of Expressing Uncertain Causalities as Conditional Causal Possibilities

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1 Evaluaton of Expressng Uncertan Causaltes as Condtonal Causal ossbltes Koch Yamada Department of lannng & Management Scence, agaoa Unversty of Technology eng & Regga (v u u u v v u (v u ) 0 u v V [1] [1] ( ) [2] [2] U V U eng [3] u u v condtonal causal probablty U V ( ) (1) ( ) (u ) ( ) [5] [5] U V ( )=a ( ) () (1) a=1/() () ( ) ( ) u u v ( ) 2 U 2 V [3] u v,u U,v V u u v eng [3] 2.1 U V u 9

2 U,(=...,) v (=...,M) u U v V v :u u v ( ) u ( ) ( ) u (2) u u (3) (4), u, u, v ( ) ( ) U V possblstcally ndependent [6] 2.5 w 1, w 2 w 1 { e e1} w 2 { e2, e2} w 2 w 1 π( w2 w1) = π( w2) (10) x z 2.1 (1) x { u, u} x Õ (2) z {, } 2.2 u { u ), u )}={ π, π } (v :u u ) eng (v :u u ) u u v (2) π( u ) ) (5) (11) (12) non- (13) z'' ) = π(( ) u z'' ) = z'' ) π( u) = u) π( u) = ) z'' ) = 1 z'' ) = 1 ) = ) nteractve)[7] π( zz '') = π( z z '') π( z '') = π( z) π( z'' ) z z ÕÕ π( u ) = π( u) u) (6) ) = π(( ) u) = π( u) π( u) (7) ( ) mn (max) (5), (6), (7) u) u) (8) 2.3 X u X 2.4 X x, z x { u, u }, z {, } z π( z x X) = π( z (9) z u u x { u, u}, y { v, v } yx1... x 1 = π x,..., x, x... x x x+..., (14) π( x1... x 1x ) = x,..., x, x... ) x,..., x (1) x,(=...,) (14) π( x1... x 1x ) π( x1... x 1x ) = π( x1)... π( x 1) π( x+ 1)... π( ) y = v y = v (1) y = v 3.1 [3] : ( u1... ur) : u1... ur, r (ont-causaton event) (15) 10

3 : u1... ur, ( ) u1... ur r (16) = : u1... ur u...,u r (3) u u v { ( ) u} ( ) (17)... ) = π( ( )... ) = π(... ) q (18)... ) = (19) (14), (15), (18), (19) = π( x) : u x) x,..., x, 1 1 x+..., (20) u), f x = u (21) u) = 0, f x = u (2) y = v ( v u v u u U : ) ( : ) (22)... ) = π( ( )... ) (23) z {, } π( ( )... ) =... ) = (24) = π( x ) : u x) x,..., x, 1 1 x+..., ( π( x) : u x) ) = x..., x π( : ) x+..., x u x = ( π( x) : u x) ) x,..., x, 1 1 x+..., (25) u), f x = u u) = f x = u (26) u) = π, f x = u, z u) = π, f x = u, z π( z u) = 0, f x = u, z : u u) = f x = u, z (27) π( x) : u x) x,..., x,, 1 1 x+..., f y = v ( π( x) : u x) ) x..., x x+...,, fy (28) u η( u ) η( u) = 1 π( u) (29) η( u) π( u) 3.2 π( z η( z 1 u) = 1 π = η, f x = u, z = v : u 1 u) = 1 π = η, f x = u, z η( z 1 u) = 0, f x = u, z 1 π( v = 1 u), f x = u, z π η (28) (30) U={u },(=...,) V={v }, (=...,M) Q + ={v 1,...,v m },m²m Q - ={v m+1,...,v },m<²m V-(Q + Q - ) Q + Q - 11

4 U ={u 1,...,u n },n² os( Q, Q ) = π( u 1... un (31) un u v1... vm vm v) π( x ) u) = π η( u) = η = 1 π π Q x y 4.1 x { u, u}, y {, } (31) + os( Q, Q ) = π( x 1... y 1... y) (32) os(, Q ) = π ( = 1, y1... ) (33) x (y x ) [8] 1., 0 f π( x) ) ), f π( x) > ) (34) ) π( x y) = π( x) (35) π( x y..., y) π( x )( π( )... π( )) π( x y... y ) y1 x y x 1 = π( y1)... π( y) (36) ) = ( π( x) ) (37) = x { u, u} (28) (y x ) os( Q +,Q - ) π( x) = 11, =, os( Q, Q ) = (38) π( y1)... π( y) (27), (28), (37) π( y ) > ), x { u, u}, π( x) > 0 (39) 2), ; u) = π 0 (40) 3), ; η( u) = 1 π 1 (41) 1) u 2) v 3) v 4.2 y y Õ π( y1... y) π( x1... y1... y) = π( x1... ) π( y1... y... ) x, y (32) ) os( Q, Q ) = = π( x)... ) = = (42) (43) ) π( x) π ( y... ) (44) = = = os( Q, Q ) π( x )... ), = = f ) > π( x )... ) = = = [ π( x )... ), 1], = = f ) = π( x )... ) = = =... ) (18), (19) (23), (24) π(v, f y =... ), y = v f = (38) os( Q +,Q - ) { π, π} u) η( u) (46) Case-I,II, III {u 1,u 2,u 3 } u π( u ) = ( = 2, 3) π( u ) 1.0 Case- 0.1 Case-) Case- π( u ) π( u ) os( Q +,Q - ) π( u ) π(v u) os( Q +,Q - ) π( u ) π( u ) Case- Case- (28) (30) (a), (b) Case- 12

5 u) π(v u ) = 0 u,v u) Case- u v (38) os( Q +,Q - ) Case- os( Q +,Q - ) (a),(b) Case- Case-I,II, III (38) Case-I II {u 1 } {u 2 } (38) {u 3 } Case-III v 3 v u 1 u 2 Case-I {u 3 } Case- Case-I ={u 3 } π( u ) π( v3 u3 ) (38) u 3 {u 1,u 3 } {u 2,u 3 } u 1 v 1 {u 1,u 3 } Case- (38) Case-I {u 3 } (17) v u (18),(19),(21) u1... u) = 0 (47) y π( y1... y u1... u) = u1... u) = (48) x = u y 0 ={u 3 } u u 3 v 1 v 3 0 (38) 0 (14) u (47) u) 13

6 (38) non-nteracton, Fuzzy Sets and Systems pp (1978) [7].. Zadeh: Fuzzy sets as a bass for a theory of possblty, Fuzzy Sets and Systems pp.3-28 (1978) [8] M. Toga: fuzzy nverse relaton based on Gšdelan logc and ts applcatons, Fuzzy Sets and Systems, 17, pp (1985) 6 2 U V u,(46) V V 2 (EDO) [1] J. earl : robablstc Reasonng n Intellgent Systems: etwors of lausble Inference, Morgan Kaufmann ublshers, Inc. (1988) [2] Y. Iwasa : Real-World pplcatons of Qualtatve Reasonng, IEEE Expert, Vol. 12, o. 3, pp (1997) [3] Y. eng, J.. Regga : bductve Inference Models for Dagnostc roblem-solvng, Sprnger-Verlag (1990) Vol..8,o.3, pp (1996) [5] K. Yamada, M. Honda : Method of Dagnoss Usng ossblty Theory, The nth Internatonal Conference on Industral & Engneerng pplcatons of rtfcal Intellgence & Expert Systems (IE-IE 96), pp , Fuuoa (1996) [6] E. Hsdal: Condtonal ossbltes Independence and (44) (28) π( x) π( x ) : u x) π x,..., x, (x ), 1 1 x+..., f y = v ( π( x) : u x) ) x..., x x+..., π(x ), f y, π( x) : u x) x x..., x+..., f y ( π( x) : u x) ), x..., x x+..., f y ) = ( π( x) ) = x { u, u} π( x ) : u x) x,..., x, 1 f y ( π( x) π( : u x) ), x..., f y π( x)... ) = = = π( x)... ) = = (18),(19) (23),(24) { } π( x)... ) = x u x y = π( ) π( : ), f = x u x y = π( ) π( : ), f = ) π( x)... ) = (44) 14

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