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Chapter 4




      BELIEF UPDATING BY NETWORK
              PROPAGATION

  Probabilistic Reasoning In Intelligent
                 Systems.
•
    – 4.1
•             -   -
    – 4.2.1
    – 4.2.2
    – 4.2.3
    – 4.3             (Polytrees)
確率伝播
1.
     i.
     ii.

2.

           1.i.
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
    –
    –
•


•
          →




    =1?
•


•


    –

    –
        OK
•
    –
•
    –
    –
•                   N1            N1

    –
    – N1
•
    –
               t2        t3


                              →
                              …




           →
確率伝播
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)
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)
             •
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     
                                                                    
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  
                                                                                                    
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
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)
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
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
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)
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       
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’
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)
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
• 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

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確率伝播

  • 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