Gouws Python 2010

8/10/2019 Gouws Python 2010

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A Python implementation of graphical models

by

Almero Gouws

March 2010

Thesis presented in partial fulfilment of the requirements for the degree

of Master of Science in Engineering at Stellenbosch University

Supervisor: Professor Ben Herbst

Department of Applied Mathematics, Faculty of Engineering


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x6

x5

x4

x3 x2

p(x1)

p(x2)

p(x3) p(x2)


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n


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p(x1)

p(x2|x1)

1(x1, x2)

2(x2, x3)

1(x1, x2) 1(x1, x2, x3)

2(x2, x3) 2(x1, x2, x3)

3(x1, x2, x3) 1(x1, x2, x3) 2(x1, x2, x3)

E3(x1, x2)

3(x1, x2, x3)

x3= 0

E3(x1, x3) 3(x1, x2, x3) x2= 1

1 x1 x2

p(x1) p(x2)

1 2

2x2(x2)

[1(x1, x2) + x21(x2)]

1x1(x1)

p(x1)

p(x2|x1)

(x1, x2) ES S(x1, x2)

ES S(x1, x1)

E(x1|x2= 1)

M(x1|x2= 1)

R

(x1, x2)


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N

x = {x1,...,xN} x

x N x = {x1,...,xN}

N

x

x

xi

{x1, x2, x3, x4, x5, x6}

x2

x1 x2 p(x2|x1)


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xi

p(xi|i) i xi

N

p(x) =N

i=1

p(xi|i),

i xi

p(x) =p(x1)p(x2|x1)p(x3|x1)p(x4|x2)p(x5|x3)p(x6|x2, x5).

xi

xi

x1

p(x1)


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x1 p(x1)

x1 = 0 x1= 10.4 0.6

p(x1)

x2 x1 p(x2|x1)

x2

x1 p(x2|x1)

x1= 0 x1 = 1x2 = 0 0.7 0.6x2 = 1 0.3 0.4

p(x2|x1)


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N x = {x1,...,xN} x

N

x = {x1,...,xN} N x

{x1, x2, x3, x4, x5, x6}

x2 x5 x6

x2 x5

x6 x2 x5

x6

xs (xs) C

C

p(x) = 1Z

cC

c(xc),


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xc c 1

Z

Z=x

cC

c(xc).

p(x) = 1

Z(x1, x2)(x1, x3)(x2, x4)(x2, x5, x6).

(xs)

xs

xs

{x1, x2} {x2, x3}

1(x1, x2) 2(x2, x3)

1(x1, x2) 2(x2, x3)

p(x) = 1

Z

(x1, x2)(x2, x3).


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x2= 0 x2 = 1

x1 = 0 0.8 0.1x1 = 1 0.4 0.9

1(x1, x2)

x3= 0 x3 = 1x2 = 0 0.7 0.5x2 = 1 0.5 0.3

2(x2, x3)

1(x1, x2) 2(x2, x3)

1(x1, x2)

2(x2, x3) 1(x1, x2)

2(x2, x3) 3(x1, x2, x3)

3(x1, x2, x3) =(x1, x2) +(x2, x3).

3(x1, x2, x3)

1(x1, x2) 2(x2, x3)

x1 x2 x3 1(x1, x2) x3

1(x1, x2) 1(x1, x2, x3)

1(x1, x2, x3 = 0) = 1(x1, x2, x3= 1) =1(x1, x2).

2(x2, x3)

x1 2(x1, x2, x3)

2(x2, x3, x1 = 0) = 2(x2, x3, x1= 1) =2(x2, x3).

1(x1, x2)

1(x1, x2, x3)

2(x2, x3) 2(x1, x2, x3)

x3 = 0 x2= 0 x2 = 1 x3 = 1 x2= 0 x2 = 1x1 = 0 0.8 0.1 x1 = 0 0.8 0.1x1 = 1 0.4 0.9 x1 = 1 0.4 0.9

1(x1, x2) 1(x1, x2, x3)

3(x1, x2, x3) 1(x1, x2, x3) 2(x1, x2, x3)

1(x1, x2, x3) 2(x1, x2, x3)


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x3 = 0 x2= 0 x2 = 1 x3 = 1 x2= 0 x2 = 1

x1 = 0 0.7 0.5 x1 = 0 0.5 0.3x1 = 1 0.7 0.5 x1 = 1 0.5 0.3

2(x2, x3) 2(x1, x2, x3)

x3 = 0 x2= 0 x2 = 1 x3 = 1 x2= 0 x2 = 1x1 = 0 1.5 0.6 x1 = 0 1.3 0.4x1 = 1 1.1 1.4 x1 = 1 0.9 1.2

3(x1, x2, x3) 1(x1, x2, x3)

2(x1, x2, x3)

3(x1, x2, x3) x3 = 0

3(x1, x2, x3)

x3 = 0 E3(x1, x2)

3(x1, x2, x3) x3= 0 E3(x1, x2)

E3(x1, x3) 3(x1, x2, x3) x2 = 1

x2= 0 x2 = 1x1 = 0 1.5 0.6x1 = 1 1.1 1.4

E3(x1, x2) 3(x1, x2, x3) x3= 0

x3= 0 x3 = 1x1 = 0 0.6 0.4x1 = 1 1.4 1.2

E3(x1, x3) 3(x1, x2, x3) x2= 1

p(x)


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p(x) = 1

Z1(x1, x3)2(x2, x3)3(x3, x4)4(x3, x5).

1(x1, x3) x1 x3


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x2 x3 x4 x5

p(x2|x1) p(x3|x1) p(x4|x2) p(x5|x3)

p(x2|x1) x1 x2 x1 x2

p(x2|x1)

(xc, xp) =p(xc|xp).

x6 x6 x2 x5

p(x6|x2, x5) x2 x5

x6

x2 x5 x2

x5

p(x6|x2, x5)


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(xc,xp) =p(xc|xp).

xc xp xc

x1 p(x1) p(x1)

(xc) =p(xc).


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N

p(xn) xn

p(x) = 1

Z(x1, x2)(x1, x3)(x2, x4)(x3, x5)(x2, x5, x6).

p(x1, x2, x3, x4, x5)

p(x1, x2, x3, x4, x5) =x6

1

Z

(x1, x2)(x1, x3)(x2, x4)(x3, x5)(x2, x5, x6).

k

k6

O(k6)

k6

x6

x6

p(x) = 1

Z(x1, x2)(x1, x3)(x2, x4)(x3, x5)

x6

(x2, x5, x6).

(x2, x5, x6) k3

O(k3)

O(k6)

p(x1)

p(x1) = 1

Z

x2,x3,x4,x5,x6

(x1, x2)(x1, x3)(x2, x4)(x3, x5)(x2, x5, x6).


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p(x1) = 1Z

x2

(x1, x2)x3

(x1, x3)x4

(x2, x4)x5

(x3, x5)x6

(x2, x5, x6).

m6(x2, x5) =

x6(x2, x4, x6)

p(x1) = 1

Z

x2

(x1, x2)x3

(x1, x3)x4

(x2, x4)x5

(x3, x5)m6(x2, x5),

m6(x2, x5) (x2, x5, x6) x6

x2 x5 x6

x6

x6

x6

m5(x2, x3) =

x5

(x3, x5)m6(x2, x5)

p(x1) = 1

Z

x2

(x1, x2)x3

(x1, x3)m5(x2, x3)x4

(x2, x4).

x5

m5(x2, x3) x2 x3

x2

x3

x5 x2 x3

x5


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p(x1) = 1

Z

x2

(x1, x2)m4(x2)x3

(x1, x3)m5(x2, x3),

= 1

Z

x2

(x1, x2)m4(x2)m3(x1, x2),

= 1

Zm2(x1),

x4 x3 x2

x1

x4

x3 x2

p(x1|x6)

x6 x1

x6

E(x2, x5) E

p(x1|x6) = 1

Z

x2

(x1, x2)x3

(x1, x3)x4

(x2, x4)x5

(x3, x5)E(x2, x5).

x6 x6

p(x1)


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p(x1) = 1

Z

x2

(x1, x2)x3

(x1, x3)x4

(x2, x4)x6

x5

(x3, x5)(x2, x5, x6),

= 1

Z

x2

(x1, x2)x3

(x1, x3)x4

(x2, x4)x6

m5(x2, x3, x6),

= 1

Z

x2

(x1, x2)m4(x2)x3

(x1, x3)m6(x2, x3),

= 1Z

x2

(x1, x2)m4(x2)m3(x1, x2),

= 1

Zm2(x1).

x6

(x2, x5, x6) k3

O(k3)

x5 (x2, x5, x6)(x3, x5)

k4

O(k4)

p(x1)

p(x1) p(x2)

p(x1) p(x2)

p(x2)

p(x1)


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p(x1)

p(x2)


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p(x) = 1

Z(x1, x2)(x2, x3)(x3, x4)(x4, x5)(x5, x6),

= 1

Z

5i=1

(xi, xi+1),

1Z

p(x1)

(6, 5, 4, 3, 2, 1)

x1

p(x3)

x3

(6, 5, 4, 2, 1, 3)

p(x3) = 1

Z

x1,x2,x4,x5,x6

(x1, x2)(x2, x3)(x3, x4)(x4, x5)(x5, x6),

= 1

Z

x1,x2

(x1, x2)(x2, x3)x4

(x3, x4)x5

(x4, x5)x6

(x5, x6).

x2 (x1, x2)(x2, x3) O(k3)

x1

x2

(6, 5, 4, 1, 2, 3)


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p(x3) = 1Z

x2

(x2, x3)x1

(x1, x2)x4

(x3, x4)x5

(x4, x5)x6

(x5, x6),

O(k2)

p(x3) = 1

Z

x2,x1

1i=2

(xi, xi+1)

x4,x5,x6

5i=3

(xi, xi+1)

.

x3

p(x3) = 1

Z(x3)(x3),

(x3) = x2,x11

i=2 (xi, xi+1) (x3) = x4,x5,x65

i=3 (xi, xi+1).

(x3) x2 x3 (x3)

x4 x3


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(x3)

(x3) =x2

(x2, x3)x1

(x1, x2),

=x2

(x2, x3)(x2).

(xn)

(xn1) xn xn1

xn1 x1 (x2) =

x1 (x1, x2) (xn)

(x3) =

x4,x5,x6

5i=3

(xi, xi+1),

=x4

(x3, x4)x5,x6

5i=4

(xi, xi+1),

=x4

(x3, x4)(x4).

(xn) (xn+1)

xn xn+1 xn+1

x6 (x5) =

x6

(x5, x6)

p(x1)

x1


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xn xm

xm

x3 x1

x2 x4 x5

x1

x1

x4

x3

p(x1)


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p(x1)


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3(x2) =x3

2(x2, x3),

4(x2) =x4

3(x2, x4),

2(x1) =x2

1(x1, x2)3(x2)4(x2).

x2 x1

x2

1 x21(x2)

x21(x2)

x2 3(x2) 4(x2)

x21(x2) =3(x2)4(x2).

1 x1 1

x21(x2) 1x1(x1)


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x1

1x1(x1) 2(x1)

1x1(x1) = 2(x1),

=x2

1(x1, x2)3(x2)4(x2),

1x1(x1) =x2

1(x1, x2)1x1(x1).

1 x1

1x1(x1)

x1 x2

xm s

xms(xm) =

rne(xm)\s

rxm(xm),

r ne(xm)\s xm s

s xm

sxm(xm) =xs\xm

s(xs)

rxs\xm

xrs(xr),

xs\xm s xm

x(x) = 1,

x(x) =(x).


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p(x3)

x3


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p(x2)

x2

p(x3)

p(x2)

p(x3) p(x2)

x32 2x2

x2 2 p(x3) p(x3)

x3

p(xn) xn

N


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2(N1)

(N1)

2(N1)

N

N(N 1)

p(x)

x

1


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x2= 0 x2 = 1

x1 = 0 0.4 0.6x1 = 1 0.4 0.1

1 x1 x2

p(xn) xn

p(x1) p(x2)

p(x1= 0) p(x1= 1) p(x2 = 0) p(x2 = 1)

p(x1) p(x2)

x1 = 0 x2= 0

x1 = 0 x2 = 1

xmax = argmax

x

p(x),

p(xmax) = maxx

p(x).

p(xmax)

p(x) = 1

Z1(x1, x2)2(x2, x3),

p(xmax) = 1

Zmax

x1(x1, x2)2(x2, x3).


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p(xmax) = 1

Zmax

x1max

x2max

x31(x1, x2)2(x2, x3).

max(ab, ac) =a max(b, c), a >0.

(x1, x2) x3

p(xmax) = 1

Zmax

x1max

x21(x1, x2)max

x32(x2, x3).

log(p(xmax)) = maxx

log(p(x)).

log(p(x)) = log(1

Z) + log(1(x1, x2)) + log(2(x2, x3)).

max(a+ b, a + c) =a+ max(b, c),

log(p(xmax)) = log(1Z

) + maxx1

maxx2

log 1(x1, x2) + maxx3

log 2(x2, x3).

xm s

xms(xm) =

rne(xm)\srxm(xm),

r ne(xm)\s xm s

s xm


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sxm(xm) = maxxs\xm

[log(s(xs)) +

rxs\xm

xrs(xr)],

xs\xm s xm

x(x) = 0,

x(x) = log((x)).

p(x)

p(x)


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x1

x1max = argmax

x1

1x1(x1).

1 x2= 0 x2 = 1 2 x3= 0 x3 = 1x1 = 0 x2 = 0 x1 = 1 x2 = 1

1 2

x1 x3

x32(x3) = 0.

2 x2

2x2(x2) = maxx3

[2(x2, x3) +x32(x3)],

= maxx3

2(x2, x3).

x2 = 0 x2= 1

2x2(x2)

x3

x2 = 0 x2 = 1 2

x2 = 0 x3 = 0

x2

x3

2

x2 = 1 x3 = 1


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x2 x3

x2 x2

1 1

2

x21(x2) =2x2(x2).

1 x1

1x1(x1) = maxx2

[1(x1, x2) +x21(x2)].

[1(x1, x2) + x21(x2)]

x2= 0 x2 = 1x1 = 0 x1 = 1

[1(x1, x2) + x21(x2)]

1x1(x1)

x1 = 0 x1= 1

1x1(x1)

x1 = 0 x2 = 1

x1 = 1 x2 = 0


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x1

x1 = 0 x2 = 1 x2 = 1 x3 = 1 x1 = 1

x2= 0 x2= 0 x3 = 0

s xm

xm

sxm(xm) = argmaxxs\xm

[log(s(xs)) + rxs\xm

xrs(xr)].

x5

x2 x3

x1 x2 x3 x4 x1

x2 x4


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{x1, x2, x3} {x2, x3, x5} {x2, x5, x6}

{x2, x4}

Ci

p(x) = 1

Z1(x1, x2, x3)2(x2, x3, x5)3(x2, x4)4(x2, x5, x6),

Si

Si i(Xs) Xs


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p(x) =

i i(Xi)i j(xj)

.

1Z

Z

Z

Ci Cj CiCj x5

x5

x5


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p(x1)

C1


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Cs Sm

CsSm(xm) =xs\xm

s(xs)

rne(Cs)\Sm

SrCs(xr),

xm m xs

s xs\xm xs xm ne(Cs)\Sm

Cs Sm

s s s

s(xs) =s(xs)

rne(Cs)\Sm

SrCs(xr),

Cs Sm

CsSm(xm) = xs\xms(xs).


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Sm Cs

SmCs(xm) =m(xm)

rne(Sm)\Cs

CrSm(xr),

xm m xr r ne(Sm)\Cs Sm Cs

m

Sm m m

m(xm) =m(xm)

rne(Sm)\Cs

CrSm(xr).

Sm Cs

SmCs(xm) =m(xm).

s(xs) =s(xs)

rne(Cs)\Sm

r(xr),

Cs

s

Sm

Cs

s (xs) =s(xs)

rne(Cs)\Sm

r (xr)rne(Cs)\Sm

r(xr).


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s s

Cs

s(xs) =s(xs)

rne(Cs)\Sm

r(xr)rne(Cs)\Sm

r(xr).

rne(Cs)\Sm

r(xr)

Cs Sm Sm

p(xn) xn

xn

xn p(x2)

p(x2) = 1

Z 2(x2),

Z=x2

2(x2).

p(x) =

i

i (xi)

i i (xi)

.

C4 x5

C3 C4 C3

C2 C3 C2 x5

C4

C3

x5


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xi


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Cs Sm

CsSm(xm) = maxxs\xm

log(s(xs)) +

rne(Cs)\Sm

SrCs(xr),

xm m xs s xs\xm xs xm ne(Cs)\Sm Cs Sm

Sm Cs

SmCs(xm) =m(xm) +

rne(Sm)\Cs

CrSm(xr),

xm m xr

r ne(Sm)\Cs Sm Cs

m

log(1) = 0

C1

C2 S1


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C2S1(x2, x3) = maxx4

log(2(x2, x3, x4)).

x2

x3

x4

x2 x3

C2S1

S1 C2

S1C1(x2, x3) =C2S1(x2, x3),

C1

(x1max

, x2max

, x3max

) = argmaxx1,x2,x3 [log(1(x1, x2, x3)) +S1C1(x2, x3)].

x4

x2

max

x3max

x4

CsSm(xm) = argmaxxs\xm

log(s(xs)) +

rne(Cs)\Sm

SrCs(xr),


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p(x1)


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C1 x1 x2

n(xn) xn

xn

xn

xm

Cs m(xm)


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n

xmCs(xm) =m(xm).

s(xs) =s(xs)

mne(Cs)

m(xm),

s

(xs) s(xs) ne(Cs)

Cs

Cs

xm

xm

Csxm(xm) =xs\xm

s(xs).

Cs

Cs Csxm(xm)

Cs

s

xm

m(xm) =m(xm)

Csxm(xm)

Csxm(xm).

2 1

3


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p(xn) xn xn

xn


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N

S N

S


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x1 x2

p(x1) p(x2|x1)

p(x1) x1

x1 S p(x1)

x1

x1

x1 {0, 0, 1, 0} x1 = 0 x1 = 1 p(x1 = 0) = 3/4

p(x1 = 1) = 1/4 p(x1)

x1 = 0 x1= 10.75 0.25

p(x1)

p(x2|x1)

p(x2|x1) p(x1)

x1 x2

(x1, x2) {(0, 0), (0, 1), (1, 1), (0, 0)}

p(x2 = 0|x1 = 0) = 2/4 p(x2= 0|x1= 1) = 1/4 p(x2= 1|x1= 1) = 1/4 p(x2= 1|x1= 0) = 0/4

p(x2|x1)

x1= 0 x1 = 1x2 = 0 0.66 0x2 = 1 0.33 1

p(x2|x1)

p(xi|i)


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p(xi|i) p(xi|i)

xi i

(xs)

xs

S

N

S

x1 x2


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(x1, x2)

(x1, x2)

ES S(x1, x2)

(x1, x2) x1= 0 x1 = 1 ES S(x1, x2) x1= 0 x1 = 1x2 = 0 x2 = 0 x2 = 1 x2 = 1

(x1, x2) ES S(x1, x2)

{0, 0}

{1, 1}

{0, 0}

(x1, x2)

ES S(x1, x2) x1= 0 x1= 1x2 = 0 x2 = 1

ES S(x1, x1)

(Hidden, 1)

(x1, x2)

(x1, x2)

E(x1|x2 = 1)

E(x1|x2 = 0) x1 = 0 x1= 1x2 = 1

E(x1|x2= 1)


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x1

E(x1|x2 = 1) M(x1|x2 = 1)

M(x1|x2 = 0) x1 = 0 x1= 1x2 = 1

M(x1|x2 = 1)

M(x1|x2 = 0)

(x1, x2)

R

(x1, x2)

R(x1, x2) x1= 0 x1 = 1x2 = 0 x2 = 1

R(x1, x2)

R(x1, x2) ES S(x1, x2)

ES S(x1, x2) x1= 0 x1= 1x2 = 0 x2 = 1

(x1, x2)

(x1, x2) x1= 0 x1 = 1 ES S(x1, x2) x1= 0 x1 = 1x2 = 0 x2 = 0 x2 = 1 x2 = 1

(x1, x2) ES S(x1, x2)

(x1, x2)


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(Xs)

(Xs)

(Xs) ES S(Xs)

Sj S

ES S(Xs)

Sj

(Xs)

ES S(Xs)

ES S(Xs)

ES S(Xs) (Xs)

ES S(Xs)


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i(Xs)

ES Si(Xs)

Sj S

Sj

i(Xs)

i(Xs)

ES Si(Xs)

i(Xs)

i(Xs)

ES Si(Xs)

ES Si(Xs)

i(Xs)

ES Si(Xs) i(Xs)


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ES Si(Xs)


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{xn, yn}


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xi {0, 2, 5}

{x0, x2, x5}

xi

k

xi= k


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xi

k

xi= k

mlc[i] =k

xi k


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N

N

N

N

i

j


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x1 x2 x3 x4 x5 x6

x1 x2 x3 x4 x5 x6

xi xi

{x1 : 2, x2 : 4, x3 : 3, x4 : 5, x5 : 2, x6 : 10} [2, 4, 3, 5, 2, 10]

x4

xi xj

xi k


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xi


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k

xi = k xi

104


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232


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k

xi = k xi


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N

xi xj 1

x1 x2 x3 x4 x5 x6x1 x2 x3 x4 x5 x6

xi


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xi xj

xi k

xi


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k


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xi = k xi

104

232


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k

xi = k xi


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xi xp

xi {xi,xp}

p(x2|x1) {x1, x2}


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{x1, x2}

x1

x2

= [x2, x5, x6]

= [2, 4, 3]

x2

x6


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xi

k

xi=k

xi


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(x1, x2, x3) x3= 1

E(x1, x2)

x3 x3 = 0

E(x1, x2) (x1, x2, x3)

Xj Xi Xi nbr nbr[j][0]

Xi Xj


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{XiXj} nbr[j][1] Xi Xj

(Xi Xj) 2(x2, x3, x5) 2(x2, x4, x5) 3(x2, x3) 4(x2, x5, x6)

2(x2, x3, x5) nbrs

nbrs[1][0] = [2, 3]

nbrs[1][1]

1(x2, x3)

nbrs[3][0] = [2]

nbrs[3][1]

2(x2)

nbrs[4][0] = [2, 5]

nbrs[4][1]

3(x2, x5)

nbrs

xj


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k

xdomain[i]=k


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xi


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xi

k xi=k

xi


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{x2, x3}

x2

x3

= [x1, x3, x4] = [3, 7, 2] x3 x4

xj


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xPOTENTIAL.domain[i] k xPOTENTIAL.domain[i]= k


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xkey

xkey


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i j

xi

xi


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xi k

xi=k


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xi k

xi=k


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xi k

xi=k


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xi k

xi=k


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xi k

xi=k


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127/139


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xi xj

xi xj


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xi xj

xi xj


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i

xi xi

yi xi

yi i

{xi, xj}

(xi, xj) =e(xixj)

2

i ,

i

{xi, yi}

(xi, yi) =e(xiyi)

2

n ,


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n


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(B1, Bt,t1) B1

Bt,t1


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Gouws Python 2010

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