- 1. Chapter 4 BELIEF UPDATING BY NETWORK PROPAGATION Probabilistic Reasoning In Intelligent Systems.
- 2. • – 4.1 • - - – 4.2.1 – 4.2.2 – 4.2.3 – 4.3 (Polytrees)
- 4. 1. i. ii. 2. 1.i.
- 5. P(xi , x j ) ∑ P(x , x i j | xk ) xk ∈X k P(xi | x j ) = = P(x j ) P(x j ) Xi • ( Xk) • – Xk – • – Xj – –
- 6. • • → =1?
- 7. • • – – OK
- 8. • – • – –
- 9. • N1 N1 – – N1 • – t2 t3 → … →
- 11. M y|x ≡ P(y | x) ≡ P(Y = y | X = x) y x P(y1 | x1 ) P(y2 | x1 ) L P(yn | x1 ) P(y1 | x2 ) P(y2 | x2 ) L P(yn | x2 ) = M M O M P(y1 | xm ) P(y2 | xm ) L P(yn | xm ) +: ¬: / +a:A ¬a:A e X BEL(x) ≡ P(x|e) f(x) λ(x) f(x)g(x) / (1,2,3)(3,2,1)=(1x3,2x2,3x1)=(3,4,3) f(x) g(x) / (1,2,3) (3,2,1) = 1x3+2x2+3x1 = 10 ( ) f (x)• M ≡ f (x)M y|x ∑ y|x x α ( 1 ) β 1 = (1,1,1,1)
- 12. X 3x3 3 0.80 if x = y x, y = 1, 2, 3 M y|x = P(y | x) = 0.10 if x ≠ y x, y = 1, 2, 3 Y ∑ P(z | y) = 1 for y = 1, 2, 3 z Z virtual P(z |y=1):P(z |y=2): P(z |y=3)=8:6:5 MYCIN • unity( 1) •
- 13. 2 chain X Y e={Y=y} X P(x) My|x X→Y λ(x)=P(e|x) BEL(x)=P(x|e) =P(Y=y|x) =αP(x)P(e|x) ∵ =My|x (0,...,1,...,0) =αP(x)λ(x) Y=y =[P(e)]-1P(x)λ(x) ※ y P(y | x )L P(Y = y | x ) LP(y | x ) 1 1 1 n 1 M M M P(y | x )L P(Y = y | x ) P(y | x ) 1 m m n m
- 14. 3 chain X Y Z λ(y)=P(Z=z|y) e={Z=z} BEL(x)=P(x|e) P(x) λ (x) = P(e | x) My|x =Mz|y Mz|y =αP(x)λ(x) =[P(e)]-1P(x)λ(x) = ∑ P(e | y, x)P(y | x) ←x y y = ∑ P(e | y)P(y | x) ←I(X,Y,Z) y = M y|x • λ (y) P(y | x ) L P(y | x ) λ (y ) 1 1 n 1 1 M y|x • λ (y) = M O M • M P(y | x ) L P(y | x ) λ (y ) 1 m n m n P(y | x )λ (y ) +L+ P(y | x )λ (y ) 1 1 1 n 1 n = M P(y | x )λ (y ) +L+ P(y | x )λ (y ) 1 m 1 n m n
- 15. Mu|t Mx|u My|x Mz|y P(t) T U X Y Z … Evidence e λ(t) λ(u) λ(x) λ(y) λ(z) • λ λ • λ Evidence e
- 16. e+ T U X Y Z e- π (x) = P(x | e+ ) π(x)=P(x|e+) BEL(x) λ(x)=P(e-|x) = ∑ P(x | u, e+ )P(u | e+ ) ≡P(x|e+,e-) u =αP(e-|x,e+)P(x|e+) = ∑ P(x | u)π (u) =αP(e-|x)P(x|e+) u =αλ(x)π(x) = π (u)• M x|u • π(x) λ(x) • π(x)
- 17. X π(u) π(x) π(x) Mx|u BEL(x) My|x λ( ) λ( ) λ(y) X π(u) π(x) π(x) Mx|u BEL(x) Mx|u λ( ) λ(u) λ(x) λ
- 18. U BEL(x)=P(x|e X ,e X )=αP(eX |x)P(x|eX )=αλ ( x)π ( x), + - - + λX(u) πX(u) V W λ(x) = P(eX |x)=P(eY ,eZ |x)=P(eY |x)P(eZ |x)=λY (x)λZ (x), - - - - - X λY(x) λZ(x) π(x) = P(x|eX )=∑ P(x|eX ,u)P(u|eX ) = ∑ P(x|u)P(u|eX ) + + + + u u πY(x) πZ(x) Y Z = ∑ P(x|u)π X (u) = M x|u • π X (u), u ∴ BEL(x) = αλY (x)λZ (x)∑ P(x|u)π X (u). u • X (V,U,W) e+ U • chain – X root λ – π
- 19. λX(u) πX(u) X M λ π M λ(x) π(x) π j (x) = π Y (x) = P(x | eY ) j + j = P(x | e+ , eY1,L) X − ∏λ ( x) α(λπ) = α ∏ P(eYk | x)P(x | eX ) k k − + k≠ j BEL(x) BEL(x) = α ∏ λYk (x) π (x) α k≠ j BEL(x) α λ1 (x) λ2 (x) BEL(x) λ1(x) λ2(x) π1(x) π2(x) =α λ j (x)
- 20. Polytree( U1 U2 ... Un − + 1 2 n BEL( x) = αP(e X | x) P( x | e X ) = αλ ( x)π ( x). − { Let e X = e − XY1 ,K, e − XYm } { e + X = e + U1 X , K , e + U n X , } X m λ ( x ) ≡ P (e − X | x ) = P (e − XY2 ,K, e − XYm | x) = ∏ λY j ( x), j =1 Y1 Y2 ... Ym Let π X (ui ) = P(ui | e +U i X ). π ( x) ≡ P ( x | e +U X , K , e +U X ) 1 n • = ∑ P( x | u ,K, u ) P(u ,K, u u1 ,K,u n 1 n 1 n | e + U1 X , K , e + U n X ) n = ∑ P( x | u)∏ π X (ui ) ※ X Y u i =1 m n BEL( x) = α ∏ λY j ( x) ∑ P( x | u)∏ π X (ui ) Z=z j =1 u i =1
- 21. Polytree • Polytree • 3 1. π (causal support) 2. λ (diagnostic support) 3. P(x|u1,...,un): (conditional-probability matrix) • 3step 1. Belief-updating U1 U2 ... Un X π λ P(x|u1,...,un) BEL(x)=αλ(x)π(x) 2. Bottom-up propagation X λ λ X 3. Top-down propagation X π π • ( ... Y1 Y2 Ym 1. π(x) P(x) 2. λ(x) = (1,1,…,1) 3. λ(x) = δx,x’ =(0,...,1,...,0) : x’ 1 X Z Z→X λ δx,x’
- 22. Polytree (1) 1. Belief-updating X π λ P(x|u1,...,un) BEL(x)=αλ(x)π(x) λ ( x) = ∏ λY ( x),π ( x) = j j ∑ u1 ,K,u n P( x | u1 ,K, un )∏ π X (ui ), i 2. Bottom-up propagation X λ λ λX (ui ) = β ∑ λ ( x) ∑ P( x | u1 ,K, un )∏ π X (uk ) ※tree x u k :k ≠ i k ≠i (2) 3. Top-down propagation X π π BEL( x) π Y j ( x j ) = α ∏ λYk ( x) ∑ P( x | u1 ,K, un )∏ π X (ui ) = α . k≠ j u1 ,K,un i λY j ( x)
- 23. Polytree (2) λX (ui ) = P(eU X | ui ) = P(eVX , e− | ui ) − + + + i X e Ui X e VX = ∑∑ P(eVX , e− | ui , v, x)P(v, x | ui ) x + v X (x ) = ∑∑ P(e− | x)P(eVX | v)P(v, x | ui ) X + Ui V x v ( ) + P(v | eVX ) = β ∑∑ P(e | x) − P(x | v, ui )P(v | ui ) λX (ui ) + P(v | eVX ) x v P(v) X ( ) − X = β ∑∑ P(e− | x)P(v | eVX )P(x | v, ui ) + = P(e Ui X | ui ) V Ui x X v λ (x) λ (x) P(x | v, ui ) = P(x | u) eUi X = {eVX , e− } − + ∑L = ∑L X X v uk :k≠i P(v | eVX ) = ∏ P(uk | eVX ) = ∏ π X (uk ) + + k≠i k≠i λ λX (ui ) = β ∑ λ (x) ∑ P(x | u)∏ π X (uk ) x uk :k≠i k≠i
- 24. • P(x|u) – U1,...,Un • → – disjunctive interaction ( )→noisy-OR-gate – global inhibition – enabling mechanism noisy-OR-gate U: I: U1 U1 U U I1 1 In n global inhibition enable mechanism AND AND OR OR Global … inhitibor Enabler OR AND AND AND AND X X X