Lecture BayesNets

  • View
    196

  • Download
    0

Embed Size (px)

Text of Lecture BayesNets

, - ( )

2003

2003V -

1 -

2003

004.032.26 (06) 32.8185 82 2003. V - 2003: . 1. .: , 2003. 188 .

, - , 2931 2003 V 2003. , , - .

. . ,

ISBN 5726204719

c - ( ), 2003

. . . 149 . . . . . . . . . . 150 . . . . . . . . . . . . 152 . . . . . . . . . . 152 153 . . . . . . . . . . . . . 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 . . . 166 . . . . . . . . . . . . . . . . . 167 . . . . . . . 168 . 169 . . . . . . . . . . . . . . . . . . . . . . . . 172 . 173 . . . . . . . . . . . . . . . . 175 . . . . . . . . . . . 177 . . . . . . . . . . . . . . . 180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 . . . . . . . . . . . . . 180 . . . . 181 . . . . . . . . . . . . . . . . 181 . . . . . . . . . . . . . . . . . . . . . . 181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

004.032.26 (06)

3

. . - , . ; , . , E-mail: alife@narod.ru - . , . . .

S. A. TEREKHOFF Snezhinsk Institute of Physics and Technology (SFTI), Snezhinsk; NeurOK LLC, Moscow, E-mail: alife@narod.ru BAYESIAN NETWORKS PRIMERAbstract Bayesian networks represent probabilistic graph models of casual relations between variables in statistical information modelling. Bayesian networks consolidate empirical frequencies of events, subjective beliefs and theoretical representations of mathematical probabilities. It is the important practical advantage which distinguishes Bayesian networks from other techniques of information modelling. This survey lecture presents a brief introduction to calculation of probabilities and statistical inference methods in Bayesian networks.

004.032.26 (06)

149

ISBN 5726204719

. . : , [26], . , ( ) : ( , . .); ( , , ). [5]. : , , . 10 . , , ( ). 10 , ! , . 10 . 10 . , , . . , , , 150 004.032.26 (06)

. . . , . , . , , . , . , . , - . , . . , , , . , , . . . . , . .

004.032.26 (06)

151

ISBN 5726204719

. , , , , , . : ; , ; . . 60- () [6], , . . , , . : ; ; . 80- 1 [21] .1 -, 20- . (Wright S. Correlation and causation // Journal of Agricultural Research, 20, 557, 1921).

152

004.032.26 (06)

. . . , , . . , ? , . , , , . [15]. , . 2 : , . , , , , . , . , , . , . , . , . , , . , . , , . , , : , , . , , , , , , , .2

004.032.26 (06)

153

ISBN 5726204719

() , . , (explaining away). . . . , , . , ( ), (A, B) , A B. A B, , , - A. . 1.

Rain?

Sprinkler?

Watson?

Cholmes?

. 1.

. 1 . 1 0 (/). 3 . . 1 .3 (, ). .

154

004.032.26 (06)

. . , , , . 80- , [21]. , , . , . : (. 2a), (. 2b), (. 2c). . 2c , -, , 4 . , A B C. , A , , (, ). , , B C . () d- (d-separation). (d-). A B d-, , , V , : V , V , , V , . , (. 1) ? ? d-. ?, ( ).4 ,

, .

004.032.26 (06)

155

ISBN 5726204719

A

B(a)

C

A

B

C(b)

...

E

B

C

...

E

A(c)

. 2. (a) . A C , B . (b) . A, . (c) . A , , B, C, . . . , E, . A .

156

004.032.26 (06)

. . d- -, , d- . , A C, B, P (A | B) = P (A | B, C). , , , , , , . , : , . ! , . , ( , !). , . , , , - . , .

, () . . 004.032.26 (06)

157

ISBN 5726204719

. P (A | B) = x . , B ( , A), A x. A B : P (A, B) = P (A | B) P (B) . (), (), . P (A | B) A , B. B, A . , A1 , . . . , An ( ) ( ). P (Aj | B) Aj , B, P (A j ): P (Aj | B) = P (Aj ) P (B | Aj ) . n P (B) = j=1 P (Aj ) P (B | Aj )

P (B | Aj ) (likelihood), P(B) (evidence) 5 . . . 5 evidence , . evidence , . , evidence. , evidence , . evidence -.

158

004.032.26 (06)

. . , . , , , : P (A1 , . . . , An ) =j

P Aj | pa(Aj ) .

pa(Aj ) - Aj . . , - Aj . , : ; ; ; - A - B1 , . . . , Bn P (A | B1 , . . . , Bn ). A , () P (A). () . , . , . , . (. 1): R , S 004.032.26 (06)

159

ISBN 5726204719

, C , W . 0 , (f ) 1 (t). P (R, S, C, W ), , 16 . , P (R, S, C, W ) = 1 .R={t,f },...,W ={t,f }

, . , , , , : P (R = f | W = t) =S={t,f },C={t,f }

P (R = f, W = t) = P (W = t) .

=

P (R = f, S, C, W = t) P (R, S, C, W = t)

R={t,f },S={t,f },C={t,f }

[7, . 45] : P (R, S, C, W ) = P (R) P (S | R) P (C | R, S) P (W | R, S, C) . , . : P (R, S, C, W ) = P (R) P (S) P (C | R, S) P (W | R) . , , . , . , -, . , 6 .6 , .

160

004.032.26 (06)

. . 1.

P (R = t) 0.7 R t f R t t f f

P (R = f ) 0.3 P (W = t | R) 0.8 0.1 S t f t f

P (S = t) 0.2

P (S = f ) 0.8

P (W = f | R) 0.2 0.9 P (R = f ) 0.1 0.2 0.3 0.9

P (R = t) 0.9 0.8 0.7 0.1

, , , 7 . . ( !) 16 , , . . : , . P (C = t) =R={t,f } S={t,f }

P (R) P (S) P (C = t | R, S) =

= 0.3 0.2 0.9 + 0.3 0.8 0.8 + 0.7 0.2 0.7 + 0.7 0.8 0.1 = 0.4 .7 . . , , , , .

004.032.26 (06)

161

ISBN 5726204719 , P (W = t) =

P (R) P (W = t | R) = 0.31 .R={t,f }

, , , () : P (C = t | R = t) =S={t,f }

P (S) P (C = t | R = t, S) = 0.82 ,

P (W = t | R = t) = 0.8 . , , . , , ? : P (R = t | C = t) = P (S = t | C = t) = P (R = t, C = t) 0.054 + 0.192 = = 0.615 . P = (C = t) 0.4 0.054 + 0.098 P (S = t, C = t) = = 0.38 . P = (C = t) 0.4

, , S W , - P (W | R). , 0.3 0.2 , . , , : P (R = t, C = t, W = t) = P (C = t, W = t) 0.8 0.3 (0.18 + 0.64) 0.1968 = = = 0.9274 , 0.8 (0.054 + 0.192) + 0.1 (0.098 + 0.056) 0.2122 P (S = t | C = t, W = t) = 0.2498 . P (R = t | C = t, W = t) = , . 162 004.032.26 (06)

. . , (. . ) 0.2. , . , 8 . , , . , -. 9 . , [19]. , 8 64 , . 8 , , , , . 8! . , , , . , , , . , . , 8 NP-. (Cooper G. F. The computational complexity of probabilistic inference using Bayesian belief networks // Artificial Intelligence. 1990. 42, (23). pp. 393405). 9 LHS Latin Hypercube Sampling.

004.032.26 (06)

163

ISBN 5726204719

, , , 10 . 1, 2, . . . , 8, . . N D . (N D), 1, 2, . . . , N . , N , , , , . - . [4]. .

A

B

C. 3.

, . (100 3), 100 . , , :10

.

N/2 . . . + N/2

164

004.032.26 (06)

. . 2. A P (A) B P (B) a1 0.20 a1 0.4 a2 0.35 a2 0.6 a3 0.45 a1 P (C | A, B) c1 c2 c3 b1 0.01 0.68 0.31 b2 0.04 0.93 0.03 b1 0.28 0.52 0.20 a2 b2 0.06 0.12 0.82 b1 0.18 0.50 0.32 a3 b2 0.82 0.10 0.08

3. (100 3) N 1 2 ... 39 ... 100 A 25 14 ... 47 ... 69 B 13 91 ... 32