( ) 4. 4.1 2009 1 14 ( ) ( ) 4. 2009 14.1 14 ( ) 1 / 41
1 2 3 4 5 4.1 ( ) 4. 2009 14.1 14 ( ) 2 / 41
X i (Ω) <, i N := {1,, N} P X1 X N (x 1,, x N ): x 1 X 1 (Ω),, x N X N (Ω) P Xi (x i ) := (N ) 2 : P Xi X j (x i, x j ) := x 1,,x i 1,x i+1,,x N P X1 X N (x 1,, x N ) (1) x 1,,x i 1,x i+1,,x j 1,x j+1,,x N P X1 X N (x 1,, x N ) ( ) 4. 2009 14.1 14 ( ) 3 / 41
4.1: 3 X 1 (Ω) = X 2 (Ω) = {1, 2, 3} X 0 : X 1 : X 2 : X 3 P X2 X 1 (1 1) P X2 X 1 (2 1) P X2 X 1 (3 1) P X2 X 1 ( ) := P X2 X 1 (1 2) P X2 X 1 (2 2) P X2 X 1 (3 2) P X2 X 1 (1 3) P X2 X 1 (2 3) q 11 q 12 q 13 P X2 X 1 (3 3) = q 21 q 22 q 23 q 31 q 32 q 33 X 0 X 1 X 2 X 3 ( ) 4. 2009 14.1 14 ( ) 4 / 41
4.1: X 0 = e 0 P X1 X 0 ( e 0 ) := [P X1 X 0 (1 e 0 ), P X1 X 0 (2 e 0 ), P X1 X 0 (3 e 0 )] = [p 1, p 2, p 3 ] X 0 X 0 = e 0 P X2 X 0 ( e 0 ) P X1 X 0 ( e 0 ) P X2 X 1 ( ) q 11 q 12 q 13 = [p 1, p 2, p 3 ] q 21 q 22 q 23 q 31 q 32 q 33 [ 3 3 3 ] = p i q i1, p i q i2, p i q i3 i=1 i=1 i=1 X 1 X 2 X 3 ( ) 4. 2009 14.1 14 ( ) 5 / 41
4.1: P X3 X 2 (e 3 ) := [P X3 X 2 (e 3 1), P X3 X 2 (e 3 2), P X3 X 2 (e 3 3)] [1, 1, 1] P X2 X 0,X 3 ( e 0, e 3 ) P X2 X 0 ( e 0 ) P X3 X 2 (e 3 ) [ 3 3 = p i q i1 1, p i q i2 1, i=1 i=1 3 i=1 ] p i q i3 1 X 0 X 1 X 2 X 3 X 0 = e 0 X 3 = e 3 ( ) 4. 2009 14.1 14 ( ) 6 / 41
4.1: P X3 X 2 (e 3 ) [r 1, r 2, r 3 ] P X2 X 0 X 3 ( e 0, e 3 ) P X2 X 0 ( e 0 ) P X3 X 2 (e 3 ) [ 3 3 3 ] = p i q i1 r 1, p i q i2 r 2, p i q i3 r 3 i=1 i=1 i=1 X 0 X 1 X 2 X 3 X 0 = e 0 X 3 = e 3 ( ) 4. 2009 14.1 14 ( ) 7 / 41
4.1: ( ) P X3 X 1 (e 3 ) q 11 q 12 q 13 q 21 q 22 q 23 q 31 q 32 q 33 r 1 r 2 r 3 = 3 j=1 q 1jr j 3 j=1 q 2jr j 3 j=1 q 3jr j P X1 X 0 X 3 ( e 0, e 3 ) P X3 X 1 (e 3 ) P X1 X 0 ( e 0 ) [ 3 3 3 = q 1j r j p 1, q 2j r j p 2, q 3j r j p 3 ], j=1 j=1 j=1 ( ) 4. 2009 14.1 14 ( ) 8 / 41
4.1: 1 P X1 X 0 ( e 0 ) = [p 1, p 2, p 3 ] P X1 X 0 ( e 0 ) = [p 1, p 2, p 3] P X2 X 0 ( e 0 ) = [p 1, p 2, p 3] [ 3 = p iq i1, i=1 q 11 q 12 q 13 q 21 q 22 q 23 q 31 q 32 q 33 3 p iq i2, i=1 3 ] p iq i3 X 0 X 1 X 2 X 3 X 0 = e 0 X 3 = e 3 i=1 ( ) 4. 2009 14.1 14 ( ) 9 / 41
4.1: 1 ( ) P X1 X 0 X 3 ( e 0, e 3 ) P X1 X 0 ( e 0 ) P X3 X 1 (e 3 ) [ 3 3 = p 1 q 1j r j, p 2 q 2j r j, p 3 j=1 j=1 P X2 X 0 X 3 ( e 0, e 3 ) P X2 X 0 ( e 0 ) P X3 X 2 (e 3 ) [ 3 3 = p iq i1 r 1, p iq i2 r 2, i=1 i=1 3 q 3j r j ], j=1 3 ] p iq i3 r 3 i=1 ( ) 4. 2009 1 4.114 ( ) 10 / 41
4.2: (1) X 4 = e 4 P X4 X 2 (e 4 ) := [P X4 X 2 (e 4 1), P X4 X 2 (e 4 2), P X4 X 2 (e 4 3)] [s 1, s 2, s 3 ] P X3 X 4 X 2 (e 3, e 4 ) := [P(e 3, e 4 1), P(e 3, e 4 2), P(e 3, e 4 3)] [r 1 s 1, r 2 s 2, r 3 s 3 ] (a) X 0 X 0 = e 0 P X2 X 0 X 3 X 4 ( e 0, e 3, e 4 ) [ 3 3 3 ] p iq i1 r 1 s 1, p iq i2 r 2 s 2, p iq i3 r 3 s 3 i=1 i=1 i=1 X 3 X 3 = e 3 X 1 X 2 X 4 X 4 = e 4 ( ) 4. 2009 1 4.114 ( ) 11 / 41
4.2: (2) 2 X 5 = e 5 P X5 X 2 (e 5 ) := [P X5 X 2 (e 5 1), P X5 X 2 (e 5 2), P X5 X 2 (e 5 3)] [0, 1, 0] P X3 X 4 X 5 X 2 (e 3, e 4, e 5 ) [r 1 s 1 0, r 2 s 2 1, r 3 s 3 0] (b) X 0 X 0 = e 0 P X3 X 4 X 5 X 2 (e 3, e 4, e 5 ) = [0, 1, 0] X 3 X 3 = e 3 X 1 X 2 X 4 X 4 = e 4 X 5 X 5 = e 5 ( ) 4. 2009 1 4.114 ( ) 12 / 41
4.2: (3) 1 X 6 = e 6 P X6 X 1 (e 6 ) := [P X6 X 1 (e 6 1), P X6 X 1 (e 6 2), P X6 X 1 (e 6 3)] [t 1, t 2, t 3 ] (c) X 0 X 0 = e 0 X 1 X 2 X 3 X 3 = e 3 X 6 X 6 = e 6 ( ) 4. 2009 1 4.114 ( ) 13 / 41
4.2: (3) 1 X 6 = e 6 ( ) P X3 X 6 X 1 (e 3, e 6 ) P X3 X 1 (e 3 ) P X6 X 1 (e 6 ) [ 3 3 3 ] = q 1j r j t 1, q 2j r j t 2, q 3j r j t 3 j=1 j=1 j=1 P X1 X 0 X 3 X 6 ( e 0, e 3, e 6 ) P X1 X 0 ( e 0 ) P X3 X 6 X 1 (e 3, e 6 ) [ 3 3 3 ] = t 1 q 1j r j p 1, t 2 q 2j r j p 2, t 3 q 3j r j p 3 j=1 j=1 j=1 ( ) 4. 2009 1 4.114 ( ) 14 / 41
M: N (a) N, a M P X1 X N (x 1,, x N ) = 1 Z Z := x i,i N a M N = {1,, N} M a M f a (x i, i N (a)), (2) f a (x i, i N (a)) {(i, a) i N (a), a M} i N M(i) M {(i, a) i N (a), a M} = {(i, a) i N, a M(i)} ( ) 4. 2009 1 4.114 ( ) 15 / 41
( ) (2) (a) A B C 1 2 (b) A B C D E 1 2 (c) A B F 1 2 (d) 1 G 2 ( ) 4. 2009 1 4.114 ( ) 16 / 41
(a) P X0 X 1 X 2 X 3 (e 0, x 1, x 2, e 3 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 )P X2 X 3 (x 2 x 3 )P X3 X 2 (e 3 x 2 ) = f A (x 1 )f B (x 1, x 2 )f C (x 2 ) N = {1, 2}, M = {A, B, C}, N (A) = {1}, N (B) = {1, 2}, N (C) = {1, 2} (b) P X0 X 1 X 2 X 3 X 4 X 5 (e 0, x 1, x 2, x 3, x 4, x 5 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 )P X2 X 3 (x 2 x 3 )P X3 X 2 (e 3 x 2 ) P X4 X 2 (e 4 x 2 )P X5 X 2 (e 5 x 2 ) = f A (x 1 )f B (x 1, x 2 )f C (x 2 )f D (x 2 )f E (x 2 ) N = {1, 2}, M = {A, B, C, D, E}, N (A) = {1}, N (B) = {1, 2}, N (C) = N (D) = N (E) = {2} ( ) 4. 2009 1 4.114 ( ) 17 / 41
( ) (4) f F (x 2 ) := f C (x 2 )f D (x 2 )f E (x 2 ) P X0 X 1 X 2 X 3 X 4 X 5 (e 0, x 1, x 2, x 3, x 4, x 5 ) = f A (x 1 )f B (x 1, x 2 )f F (x 2 ) f G (x 1, x 2 ) := f A (x 1 )f B (x 1, x 2 )f F (x 2 ) P X0 X 1 X 2 X 3 X 4 X 5 (e 0, x 1, x 2, x 3, x 4, x 5 ) = f G (x 1, x 2 ) ( ) 4. 2009 1 4.114 ( ) 18 / 41
2 2 P X1 (x 1 )P X2 X 1 (x 2 x 1 )P X3 X 2 (x 3 x 2 ) = f A (x 1 )f B (x 1, x 2 )f C (x 2, x 3 ) P X1 (x 1 )P X2 X 1 (x 2 x 1 )P X3 X 1,X 2 (x 3 x 1, x 2 ) = f A (x 1 )f B (x 1, x 2 )f C (x 1, x 2, x 3 ) ( ) 4. 2009 1 4.114 ( ) 19 / 41
4.1 i N, a M n i a, m a i : X i n i a (x i ) := 1, m a i (x i ) := 1, x i X i n i a (x i ) := m c i (x i ), (5) m a i (x i ) := c M(i)\{a} x a,i X a,i f a (x a ) n j a (x j ) (6) j N (a)\{i} 4.1 n i a (x i ) := 1, m a i (x i ) := 1, x i X i (i, a) N M (5), (6) n i a (x i ), m a i (x i ) ( ) 4. 2009 1 4.114 ( ) 20 / 41
4.1 ( ) (5) {m a i (x i )} (5),(6) : m a i (x i ) := f a (x a ) x a,i X a,i j N (a)\{i} c M(j)\{a} m c j (x j ) (7) {n i a (x i )}, {m a i (x i )}, x i X i ( ) 4. 2009 1 4.114 ( ) 21 / 41
4.1 4.1 1 (5), (6) {n i a (x i )}, {m a i (x i )} b i (x i ) a N(i) b i (x i ) P Xi (x i ) m a i (x i ), (8) x i X i b i (x i ) = 1 (9) ( ) 4. 2009 1 4.114 ( ) 22 / 41
4.1 ( ) b a (x a ) f a (x a ) n i a (x i ), (10) i N(a) b a (x a ) = 1 (11) x a X a b a (x a ) P Xa (x a ) := P Xj,j N (a) (x j, j N (a)) b a (x a ) = b i (x i ) (12) x a,i X a,i {b i (x i )} i N, {b a (x a )} a M ( ) 4. 2009 1 4.114 ( ) 23 / 41
4.5 4.1 f A (x 1 ) := P X0 (e 0 )P X1 X 0 (x 1 e 0 ) f B (x 1, x 2 ) := P X2 X 1 (x 2 x 1 ) f C (x 2 ) := P X3 X 2 (e 3 x 2 ) (a) A m A 1 n 1 A m B 1 1 n 1 B B m B 2 n 2 B m C 2 2 n 2 C C ( ) 4. 2009 1 4.114 ( ) 24 / 41
4.5 ( ) n 2 B (x 2 ) = m C 2 (x 2 ), n 1 A (x 1 ) = m B 1 (x 1 ), n 1 B (x 1 ) = m A 1 (x 1 ), n 2 C (x 2 ) = m B 2 (x 2 ), m A 1 (x 1 ) = f A (x 1 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 ), m B 2 (x 2 ) = f B (x 1, x 2 )n 1 B (x 1 ) = f B (x 1, x 2 )m A 1 (x 1 ) x 1 x 1 = f A (x 1 )f B (x 1, x 2 ) x 1 = P X0 (e 0 )P X1 X 0 (x 1 e 0 )P X2 X 1 (x 2 x 1 ), x 1 ( ) 4. 2009 1 4.114 ( ) 25 / 41
4.5 ( ) m C 2 (x 2 ) = f C (x 2 ) = P X3 X 2 (e 3 x 2 ), m B 1 (x 1 ) = x 2 f B (x 1, x 2 )n 2 B (x 2 ) = x 2 f B (x 1, x 2 )m C 2 (x 1 ) = x 2 f B (x 1, x 2 )f C (x 2 ) = x 2 P X2 X 1 (x 2 x 1 )P X3 X 2 (e 3 x 2 ), b 1 (x 1 ) m A 1 (x 1 )m B 1 (x 1 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 ) P X2 X 1 (x 2 x 1 )P X3 X 2 (e 3 x 2 ) x 2 b 2 (x 2 ) m B 2 (x 2 )m C 2 (x 2 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 )P X2 X 1 (x 2 x 1 )P X3 X 2 (e 3 x 2 ) x 1 ( ) 4. 2009 1 4.114 ( ) 26 / 41
4.5 ( ) b A (x A ) f A (x 1 )n 1 A (x 1 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 ) P X2 X 1 (x 2 x 1 )P X3 X 2 (e 3 x 2 ), x 2 b B (x B ) f B (x 1, x 2 )n 1 B (x 1 )n 2 B (x 2 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 )P X2 X 1 (x 2 x 1 )P X3 X 2 (e 3 x 2 ), b C (x C ) f C (x 2 )n 2 C (x 2 ) = P X0 (e 0 )P X1 X 0 (x 1 e 0 )P X2 X 1 (x 2 x 1 )P X3 X 2 (e 3 x 2 ). x 1 ( ) 4. 2009 1 4.114 ( ) 27 / 41
4.5 ( ) 4.2 f D (x 2 ) = P X4 X 2 (e 4 x 2 ), f E (x 2 ) = P X5 X 2 (e 5 x 2 ) (b) A m A 1 n 1 A m B 1 1 n 1 B B m B 2 n 2 B n 2 D n 2 E D m C 2 2 m D 2 n 2 C m E 2 C E ( ) 4. 2009 1 4.114 ( ) 28 / 41
4.5 ( ) m D 2 (x 2 ) = f D (x 2 ) = P X4 X 2 (e 4 x 2 ), m E 2 (x 2 ) = f E (x 2 ) = P X5 X 2 (e 5 x 2 ), n 2 B (x 2 ) = m C 2 (x 2 )m D 2 (x 2 )m E 2 (x 2 )f C (x 2 )f D (x 2 )f E (x 2 ) = P X3 X 2 (e 3 x 2 )P X4 X 2 (e 4 x 2 )P X5 X 2 (e 5 x 2 ), n 2 C (x 2 ) = n 2 D (x 2 ) = n 2 E (x 2 ) = m B 2 (x 2 ), b 2 (x 2 ) m B 2 (x 2 )m C 2 (x 2 )m D 2 (x 2 )m E 2 (x 2 ) = x 1 P X0 (e 0 )P X1 X 0 (x 1 e 0 )P X2 X 1 (x 2 x 1 ) P X3 X 2 (e 3 x 2 )P X4 X 2 (e 4 x 2 )P X5 X 2 (e 5 x 2 ) ( ) 4. 2009 1 4.114 ( ) 29 / 41
4.2 4.1 4.2 : n i a (x i ) := 1, m a i (x i ) := 1, x i X i 4.2 (i, a) N M 1 1 (i, a) (5), (6) n i a (x i ), m a i (x i ) ( ) 4. 2009 1 4.114 ( ) 30 / 41
4.2 b a = P Xa, b i = P Xi : 0 < ϵ < 0.125 P X1 X 2 X 3 (x 1, x 2, x 3 ) = f A (x 1, x 2 )f B (x 1, x 3 )f C (x 2, x 3 ), { 1 ϵ, x1 = x 2 f A (x 1, x 2 ) = ϵ, x 1 x 2, { 1 ϵ, x1 = x 3 f B (x 1, x 3 ) = ϵ, x 1 x 3, { ϵ, x2 = x 3 f C (x 2, x 3 ) = 1 ϵ, x 2 x 3 ( ) 4. 2009 1 4.114 ( ) 31 / 41
4.2 ( ) b A (x 1, x 2 ) = P X1 X 2 (x 1, x 2 ) b B (x 1, x 3 ) = P X1 X 3 (x 1, x 3 ) b C (x 2, x 3 ) = P X2 X 3 (x 2, x 3 ) 1 A B 2 C 3 ( ) 4. 2009 1 4.114 ( ) 32 / 41
4.2 ( ) m A 1 (x 1 ) = m B 1 (x 1 ) = 1, m A 2 (x 2 ) = m C 2 (x 2 ) = 1, m B 3 (x 3 ) = m C 3 (x 3 ) = 1, x 1 X 1, x 2 X 2, x 3 X 3 m A 1 (x 1 ) = x 2 X 2 f A (x 1, x 2 )m C 2 (x 2 ) = 1 a = A, B, C, i = 1, 2, 3 m a i (x i ) b i (x i ) = 1, i = 1, 2, 3 2 b A (x 1, x 2 ) = P X1 X 2 (x 1, x 2 ) b B (x 1, x 3 ) = P X1 X 3 (x 1, x 3 ) b C (x 2, x 3 ) = P X2 X 3 (x 2, x 3 ) ( ) 4. 2009 1 4.114 ( ) 33 / 41
4.2 ( ) b A (x 1, x 2 ) = f A (x 1, x 2 )m B 1 (x 1 )m C 2 = f A (x 1, x 2 ), (14) b B (x 1, x 3 ) = f B (x 1, x 3 )m A 1 (x 1 )m C 3 = f B (x 1, x 3 ), (15) b C (x 2, x 3 ) = f C (x 2, x 3 )m A 2 (x 2 )m B 3 = f C (x 2, x 3 ) (16) P X1 X 2 X 3 (0, 1, 0) + P X1 X 2 X 3 (0, 1, 1) = ϵ, P X1 X 2 X 3 (1, 0, 0) + P X1 X 2 X 3 (1, 0, 1) = ϵ, P X1 X 2 X 3 (0, 0, 1) + P X1 X 2 X 3 (0, 1, 1) = ϵ, P X1 X 2 X 3 (1, 0, 0) + P X1 X 2 X 3 (1, 1, 0) = ϵ, P X1 X 2 X 3 (0, 0, 0) + P X1 X 2 X 3 (1, 0, 0) = ϵ, P X1 X 2 X 3 (0, 1, 1) + P X1 X 2 X 3 (1, 1, 1) = ϵ P X1 X 2 X 3 (x 1, x 2, x 3 ) ϵ < 0.125 x 1 X 1,x 2 X 2,x 3 X 3 P X1 X 2 X 3 (x 1, x 2, x 3 ) < 1 ( ) 4. 2009 1 4.114 ( ) 34 / 41
P X1 X N (x 1,, x N ) := x 1 X 1,, x N X N max P X1 X x 1,,x N (x 1,, x N ) N 4.3 n i a(x i ) := 1, m a i(x i ) := 1, x i X i n i a(x i ) := m c i(x i ), (17) m a i(x i ) := c M(i)\{a} max f a (x a ) x a,i X a,i j N (a)\{i} n j a(x j ) (18) n i a (x i) := 1, m a i (x i) := 1, x i X i (i, a) N M (17), (18) n i a (x i), m a i (x i) ( ) 4. 2009 1 4.114 ( ) 35 / 41
4.3 4.3 4.3 (17), (18) {n i a (x i)}, {m a i (x i)} b i(x i ) x i X i P X i (x i ) := x i X i a N(i) b a(x a ) f a (x a ) x a X a P X a (x a ) := x a X a m a i(x i ) (19) max P X1 X x 1,,x i 1,x i+1,,x N (x 1,, x N ) N i N(a) n i a(x i ) (20) max x j, j N (a) P X 1 X N (x 1,, x N ) ( ) 4. 2009 1 4.114 ( ) 36 / 41
4.1 4.1 (1) d a,i : i N a M(i) (a ) M a,i : a (a ) N a,i : a Y a,i := j N a,i X j d a,i m a i (x i ) = f c (x c ) (13) y a,i Y a,i c M[a,i] m a i (x i ) ( ) 4. 2009 1 4.114 ( ) 37 / 41
4.1 4.1 (2) 1 d a,i = 1 (12) m a i (x i ) = f a (x i ) (13) 2 d a,i 1 = max max d c j j N (a)\{i} c M(j)\{a} (7) j N (a)\{i}, c M(j)\{a} m c j (x j ) = y c,j Y c,j f e (x e ) e M c,j ( ) 4. 2009 1 4.114 ( ) 38 / 41
4.1 4.1 (3) 3 (7) ( d a,i ) m a i (x i ) = f a (x a ) x a,i X a,i j N (a)\{i} = f a (x a ) x j X j,j N (a)\{i} = f a (x a ) y a,i Y a,i = y a,i Y a,i c M(j)\{a} y c,j Y c,j j N (a)\{i} c M(j)\{a} y c,j Y c,j j N (a)\{i} c M(j)\{a} c M a,i f c (x c ) e M c,j f e (x e ) f e (x e ) e M c,j f e (x e ) e M c,j max max d a i (13) {m a i (x i )} i N a M(i) ( ) 4. 2009 1 4.114 ( ) 39 / 41
4.1 4.1 (4) b i (x i ) = = = m a i (x i ) a M(i) a M(i) y a,i Y a,i x j X j,i j a M(i) x j X j,i j a M f c (x c ) c M a,i f c (x c ) c M a,i f a (x a ). ( ) 4. 2009 1 4.114 ( ) 40 / 41
4.1 4.1 (5) b a (x a ) f a (x a ) n i a (x i ) i N (a) = f a (x a ) = f a (x a ) = = x i X i,i N (a) i N (a) c M(i)\{a} m i c (x j ) i N (a) c M(i)\{a} y c,i Y c,i f a (x a ) x i X i,i N (a) a M i N (a) c M(i)\{a} f a (x a ). f e (x e ) e M c,i f e (x e ) e M c,i ( ) 4. 2009 1 4.114 ( ) 41 / 41