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Re Reas ason onin ing g Un Unde der Un Uncerta tain inty ty: Varia Va iabl ble eli limin inat atio ion Com omputer Science c cpsc sc322, Lecture 3 30 (Te Text xtboo ook k Chpt 6.4) June, 20 20, 2 2017 CPSC 322,

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SLIDE 1

CPSC 322, Lecture 30 Slide 1

Re Reas ason

  • nin

ing g Un Unde der Un Uncerta tain inty ty: Va Varia iabl ble eli limin inat atio ion

Com

  • mputer Science c

cpsc sc322, Lecture 3 30 (Te Text xtboo

  • ok

k Chpt 6.4)

June, 20 20, 2 2017

slide-2
SLIDE 2

CPSC 322, Lecture 30 Slide 2

Lectu ture re Ov Overv rvie iew

  • Recap

p Intr tro

  • Varia

iabl ble Eli limin inati tion

  • n
  • Variable Elimination
  • Simplifications
  • Example
  • Independence
  • Where are we?
slide-3
SLIDE 3

CPSC 322, Lecture 29 Slide 3

Bnet t Infe ference: Ge General

  • Suppose the variables of the belief network are X1,…,Xn.
  • Z is the query variable
  • Y1=v1, …, Yj=vj are the observed variables (with their values)
  • Z1, …,Zk are the remaining variables
  • What we want to compute:

) , , | (

1 1 j j

v Y v Y Z P   

           

Z j j j j j j j j j j

v Y v Y Z P v Y v Y Z P v Y v Y P v Y v Y Z P v Y v Y Z P ) , , , ( ) , , , ( ) , , ( ) , , , ( ) , , | (

1 1 1 1 1 1 1 1 1 1

    

) , , , (

1 1 j j

v Y v Y Z P   

  • We can actually compute:
slide-4
SLIDE 4

CPSC 322, Lecture 29 Slide 4

Infe ference wit ith Fa Facto tors

We can compute P(Z, Y1=v1, …,Yj=vj) by

  • expressing the joint as a factor,

f (Z, Y1…,Yj , Z1…,Zj )

  • ass

ssign gning Y1=v1, …, Yj=vj

  • and su

summing o g out the variables Z1, …,Zk

 

 

  

1 1 1

, , 1 1 1 1

) ,.., , ,.., , ( ) , , , (

Z v Y v Y k j Z j j

j j k

Z Z Y Y Z f v Y v Y Z P

 

slide-5
SLIDE 5

CPSC 322, Lecture 29 Slide 5

Varia iabl ble Eli limin inati tion

  • n Intr

tro

  • (1

(1)

  • We can express the joint factor as a product of factors
  • Using the chain r

rule and the definition

  • n of
  • f a Bn

Bnet, we can write P(X1, …, Xn) as

 n i i i pX

X P

1

) | (

 n i i i pX

X f

1

) , (

 

  

  

1 1 1

, , 1 1 1

) , ( ) , , , (

Z v Y v Y n i i i Z j j

j j k

pX X f v Y v Y Z P

 

f(Z, Y1…,Yj , Z1…,Zj )

 

 

  

1 1 1

, , 1 1 1 1

) ,.., , ,.., , ( ) , , , (

Z v Y v Y k j Z j j

j j k

Z Z Y Y Z f v Y v Y Z P

 

slide-6
SLIDE 6

CPSC 322, Lecture 29 Slide 6

Varia iabl ble Eli limin inati tion

  • n Intr

tro

  • (2

(2)

  • 1. Construct a factor for each conditional probability.
  • 2. In each factor assign the observed variables to their
  • bserved values.
  • 3. Multiply the factors
  • 4. For each of the other variables Zi ∈ {Z1, …, Zk },

sum out Zi Inference in belief networks thus reduces to computing “the sums of products….”

 

  

  

1 1 1

, , 1 1 1

) , ( ) , , , (

Z v Y v Y n i i i Z j j

j j k

pX X f v Y v Y Z P

 

slide-7
SLIDE 7

CPSC 322, Lecture 30 Slide 7

Lectu ture re Ov Overv rvie iew

  • Recap

p Intr tro

  • Varia

iabl ble Eli limin inati tion

  • n
  • Variable Elimination
  • Simplifications
  • Example
  • Independence
  • Where are we?
slide-8
SLIDE 8

CPSC 322, Lecture 30 Slide 8

Ho How to to si simpl plif ify th the Co Compu puta tati tion

  • n?

 

1

1 i)

varsX (

Z n i Z

f

k

  • Assume we have turned the CPTs into factors

and performed the assignments

) varsX ( ) , (

i

f pX X f

i i

? ) , , ( 

 t G

G D C f

 

  

1 1 1

1 , ,

) , (

Z n i v Y v Y i i Z

j j k

pX X f

Let’s focus on the basic case, for instance…

) ( ) , ( ) , , ( ) , ( D f A E f D B A f D C f

A

  

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SLIDE 9

CPSC 322, Lecture 30 Slide 9

Ho How to to si simpl plif ify: ba basi sic case se

Let’s focus on the basic case.



1

1 i)

varsX (

Z n i

f

  • How can we compute efficiently?

) ( ) , ( ) , , ( ) , ( D f A E f D B A f D C f

A

  

Factor out those terms that don't involve Z1 !

                

  

 

1 1 1

varsXi | i varsXi | i

) varsX ( ) varsX (

Z Z i Z i

f f

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SLIDE 10

CPSC 322, Lecture 30 Slide 10

Ge General l case se: Su Summin ing g out t varia iable les s effi fficie ientl tly

    

              

1 2 1

1 1 1

) (

Z Z Z h i i Z h Z

f f f f f f

k k

    

 

   

2

1

...

Z i Z

f f f

k

Now to sum out a variable Z2 from a product f1×… ×fi × f’ of factors, again partition the factors into two sets

  • F: those that
  • F: those that
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SLIDE 11

CPSC 322, Lecture 30 Slide 1 1

Analogy with “Computing sums of products”

Th This s si simplification

  • n is

s si similar to

  • what you
  • u can d

do

  • in b

basi sic alge gebra with multiplication

  • n and addition
  • n
  • It takes 14 multiplications or additions to evaluate the

expression a b + a c + a d + a e h + a f h + a g h.

  • This expression be evaluated more efficiently….
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SLIDE 12

CPSC 322, Lecture 30 Slide 12

Varia iabl ble eli limin inati tion

  • n or
  • rde

derin ing

P(G,D=t) = A,B,C, f(A,G) f(B,A) f(C,G) f(B,C) P(G,D=t) = A f(A,G) B f(B,A) C f(C,G) f(B,C) P(G,D=t) = A f(A,G) C f(C,G) B f(B,C) f(B,A) Is there only one way to simplify?

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SLIDE 13

CPSC 322, Lecture 10 Slide 13

Var aria iable le eli limi minat atio ion al algo gori rith thm: m: Su Summ mmar ary

To To c compu pute P(Z (Z| Y Y1=v =v1 ,… ,Yj=vj ) ) :

  • 1. Construct a factor for each conditional probability.
  • 2. Set the observed variables to their observed values.
  • 3. Given an elimination ordering, simplify/decompose sum of

products

  • 4. Perform products and sum out Zi
  • 5. Multiply the remaining factors (all in ? )
  • 6. Normalize: divide the resulting factor f(Z) by Z f(Z) .

P(Z (Z, , Y1…,Yj , , Z1…,Zj )

  • C. Z2
  • B. Y2
  • A. Y1=v

=v1

  • D. Z
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SLIDE 14

CPSC 322, Lecture 10 Slide 14

Var aria iable le eli limi minat atio ion al algo gori rith thm: m: Su Summ mmar ary

To To c compu pute P(Z (Z| Y Y1=v =v1 ,… ,Yj=vj ) ) :

  • 1. Construct a factor for each conditional probability.
  • 2. Set the observed variables to their observed values.
  • 3. Given an elimination ordering, simplify/decompose sum of

products

  • 4. Perform products and sum out Zi
  • 5. Multiply the remaining factors (all in ? )
  • 6. Normalize: divide the resulting factor f(Z) by Z f(Z) .

P(Z (Z, , Y1…,Yj , , Z1…,Zj )

Z

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SLIDE 15

CPSC 322, Lecture 30 Slide 15

Lectu ture re Ov Overv rvie iew

  • Re

Recap p Intr tro

  • Varia

iabl ble Eli limin inati tion

  • n
  • Variable Elimination
  • Simplifications
  • Example
  • Independence
  • Where are we?
slide-16
SLIDE 16

CPSC 322, Lecture 30 Slide 16

Var aria iable le eli limi minat atio ion exa xamp mple le

Compute P(G | H=h1 ).

  • P(G,H) = A,B,C,D,E,F,I P(A,B,C,D,E,F,G,H,I)
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SLIDE 17

CPSC 322, Lecture 30 Slide 17

Var aria iable le eli limi minat atio ion exa xamp mple le

Compute P(G | H=h1 ).

  • P(G,H) = A,B,C,D,E,F,I P(A,B,C,D,E,F,G,H,I)

Chain Rule + Conditional Independence: P(G,H) = A,B,C,D,E,F,I P(A)P(B|A)P(C)P(D|B,C)P(E|C)P(F|D)P(G|F,E)P(H|G)P(I|G)

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SLIDE 18

CPSC 322, Lecture 30 Slide 18

Var aria iable le eli limi minat atio ion exa xamp mple le (s (ste tep1)

Compute P(G | H=h1 ).

  • P(G,H) = A,B,C,D,E,F,I P(A)P(B|A)P(C)P(D|B,C)P(E|C)P(F|D)P(G|F,E)P(H|G)P(I|G)

Factorized Representation: P(G,H) = A,B,C,D,E,F,I f0(A) f1(B,A) f2(C) f3(D,B,C) f4(E,C) f5(F, D) f6(G,F,E) f7(H,G) f8(I,G)

  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
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SLIDE 19

CPSC 322, Lecture 30 Slide 19

Var aria iable le eli limi minat atio ion exa xamp mple le (s (ste tep 2)

Compute P(G | H=h1 ). Previous state: P(G,H) = A,B,C,D,E,F,I f0(A) f1(B,A) f2(C) f3(D,B,C) f4(E,C) f5(F, D) f6(G,F,E) f7(H,G) f8(I,G) Observe H : P(G,H=h1) = A,B,C,D,E,F,I f0(A) f1(B,A) f2(C) f3(D,B,C) f4(E,C) f5(F, D) f6(G,F,E) f9(G) f8(I,G)

  • f9(G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
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SLIDE 20

CPSC 322, Lecture 30 Slide 20

Var aria iable le eli limi minat atio ion exa xamp mple le (s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). Previous state: P(G,H) = A,B,C,D,E,F,I f0(A) f1(B,A) f2(C) f3(D,B,C) f4(E,C) f5(F, D) f6(G,F,E) f9(G) f8(I,G) Elimination ordering A, C, E, I, B, D, F : P(G,H=h1) = f9(G) F D f5(F, D) B I f8(I,G) E f6(G,F,E) C f2(C) f3(D,B,C) f4(E,C) A f0(A) f1(B,A)

  • f9(G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
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SLIDE 21

CPSC 322, Lecture 30 Slide 21

Var aria iable le eli limi minat atio ion exa xamp mple le(s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) F D f5(F, D) B I f8(I,G) E f6(G,F,E) C f2(C) f3(D,B,C) f4(E,C) A f0(A) f1(B,A) Eliminate A: P(G,H=h1) = f9(G) F D f5(F, D) B f10(B) I f8(I,G) E f6(G,F,E) C f2(C) f3(D,B,C) f4(E,C)

  • f9(G)
  • f10(B)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
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SLIDE 22

CPSC 322, Lecture 30 Slide 22

Var aria iable le eli limi minat atio ion exa xamp mple le(s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) F D f5(F, D) B f10(B) I f8(I,G) E f6(G,F,E) C f2(C) f3(D,B,C) f4(E,C) Eliminate C: P(G,H=h1) = f9(G) F D f5(F, D) B f10(B) I f8(I,G) E f6(G,F,E) f12(B,D,E)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
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SLIDE 23

CPSC 322, Lecture 30 Slide 23

Var aria iable le eli limi minat atio ion exa xamp mple le(s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) F D f5(F, D) B f10(B) I f8(I,G) E f6(G,F,E) f12(B,D,E) Eliminate E: P(G,H=h1) =f9(G) F D f5(F, D) B f10(B) f13(B,D,F,G) I f8(I,G)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f13(B,D,F,G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
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SLIDE 24

CPSC 322, Lecture 30 Slide 24

Var aria iable le eli limi minat atio ion exa xamp mple le(s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) F D f5(F, D) B f10(B) f13(B,D,F,G) I f8(I,G) Eliminate I: P(G,H=h1) =f9(G) f14(G) F D f5(F, D) B f10(B) f13(B,D,F,G)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f13(B,D,F,G)
  • f14(G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
slide-25
SLIDE 25

CPSC 322, Lecture 30 Slide 25

Var aria iable le eli limi minat atio ion exa xamp mple le(s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) f14(G) F D f5(F, D) B f10(B) f13(B,D,F,G) Eliminate B: P(G,H=h1) = f9(G) f14(G) F D f5(F, D) f15(D,F,G)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f13(B,D,F,G)
  • f14(G)
  • f15(D,F,G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
slide-26
SLIDE 26

CPSC 322, Lecture 30 Slide 26

Var aria iable le eli limi minat atio ion exa xamp mple le(s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) f14(G) F D f5(F, D) f15(D,F,G) Eliminate D: P(G,H=h1) =f9(G) f14(G) F f16(F, G)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f13(B,D,F,G)
  • f14(G)
  • f15(D,F,G)
  • f16(F, G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
slide-27
SLIDE 27

CPSC 322, Lecture 30 Slide 27

Var aria iable le eli limi minat atio ion exa xamp mple le(s (ste teps s 3-4) 4)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) f14(G) F f16(F, G) Eliminate F: P(G,H=h1) = f9(G) f14(G) f17(G)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f13(B,D,F,G)
  • f14(G)
  • f15(D,F,G)
  • f16(F, G)
  • f17(G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
slide-28
SLIDE 28

CPSC 322, Lecture 30 Slide 28

Var aria iable le eli limi minat atio ion exa xamp mple le (s (ste tep 5)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f9(G) f14(G) f17(G)

Multiply remaining factors: P(G,H=h1) = f18(G)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f13(B,D,F,G)
  • f14(G)
  • f15(D,F,G)
  • f16(F, G)
  • f17(G)
  • f18(G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
slide-29
SLIDE 29

CPSC 322, Lecture 30 Slide 29

Var aria iable le eli limi minat atio ion exa xamp mple le (s (ste tep 6)

Compute P(G | H=h1 ). . Elimination ordering A, C, E, I, B, D, F. Previous state: P(G,H=h1) = f18(G) Normalize:

P(G | H H=h1) ) = f18

18(G)

(G) / g ∈ dom(G) f18

18(G)

(G)

  • f9(G)
  • f10(B)
  • f12(B,D,E)
  • f13(B,D,F,G)
  • f14(G)
  • f15(D,F,G)
  • f16(F, G)
  • f17(G)
  • f18(G)
  • f0(A)
  • f1(B,A)
  • f2(C)
  • f3(D,B,C)
  • f4(E,C)
  • f5(F, D)
  • f6(G,F,E)
  • f7(H,G)
  • f8(I,G)
slide-30
SLIDE 30

CPSC 322, Lecture 30 Slide 30

Lectu ture re Ov Overv rvie iew

  • Re

Recap p Intr tro

  • Varia

iabl ble Eli limin inati tion

  • n
  • Variable Elimination
  • Simplifications
  • Example
  • Independence
  • Where are we?
slide-31
SLIDE 31

CPSC 322, Lecture 30 Slide 31

Co Comp mple lexi xity ty (n (not

  • t re

requir ired)

  • The complexity of the algorithm depends on a measure of complexity of the

network.

  • The size of a tabular representation of a factor is exponential in the number of

variables in the factor.

  • The treewidth of a network, given an elimination ordering, is the maximum number
  • f variables in a factor created by summing out a variable, given the elimination
  • rdering.
  • The treewidth of a belief network is the minimum treewidth over all elimination
  • rderings. The treewidth depends only on the graph structure and is a measure of

the sparseness of the graph.

  • The complexity of VE is exponential in the treewidth and linear in the number of

variables.

  • Finding the elimination ordering with minimum treewidth is NP-hard, but there is

some good elimination ordering heuristics.

slide-32
SLIDE 32

CPSC 322, Lecture 30 Slide 32

Var aria iable le el elim imin inat atio ion n an and co cond ndit itio iona nal l in independence

  • Variable Elimination looks incredibly painful for large graphs?
  • We used conditional independence…..
  • Can we use it to make variable elimination simpler?

Yes, all the variables from which the query is conditional independent given the observations can be pruned from the Bnet

slide-33
SLIDE 33

CPSC 322, Lecture 10 Slide 33

VE a and con

  • nditi

tion

  • nal independence: Exa

xample

  • All the variables from which the query is conditional

independent given the observations can be pruned from the Bnet

e.g .g., ., P(G | H= H=v1, F , F= v2, C=v3).

  • B. E, D
  • A. B,

B, D D, E E C . . D, I D.

  • D. B, D, A
slide-34
SLIDE 34

CPSC 322, Lecture 10 Slide 34

VE a and con

  • nditi

tion

  • nal independence: Exa

xample

  • All the variables from which the query is conditional

independent given the observations can be pruned from the Bnet

e.g .g., ., P(G | H= H=v1, F , F= v2, C=v3).

slide-35
SLIDE 35

CPSC 322, Lecture 30 Slide 35

VE a and con

  • nditi

tion

  • nal independence: Exa

xample

  • All the variables from which the query is conditional

independent given the observations can be pruned from the Bnet

e.g .g., ., P(G | H= H=v1, F , F= v2, C=v3).

slide-36
SLIDE 36

CPSC 322, Lecture 30 Slide 36

VE a and con

  • nditi

tion

  • nal independence: Exa

xample

  • All the variables from which the query is conditional

independent given the observations can be pruned from the Bnet

e.g .g., ., P(G | H= H=v1, F , F= v2, C=v3).

slide-37
SLIDE 37

CPSC 322, Lecture 4 Slide 37

Learning Goals for today’s class

Yo You c can an:

  • Carry out va

varia iabl ble e eli limin inat atio ion by using factor representation and using the factor operations.

  • Use techniques to simplify variable elimination.
slide-38
SLIDE 38

CPSC 322, Lecture 2 Slide 38

Bi Big g Pi Pict ctur ure: e: R&R &R sy syst stem ems

Environ

  • nment

Prob

  • blem

Qu Query Planning Dete terministi tic Sto tochas asti tic Se Sear arch Arc Consiste tency Sear arch Sear arch Val alue Ite terat ation Var

  • ar. Eliminat

ation Constr trai aint t Sat atisfac acti tion Logics STRIPS Belief N Nets ts Var ars + Constr trai aints ts Decision Nets ts Mar arkov Pr v Processes Var

  • ar. Eliminat

ation Sta tati tic Se Sequenti tial al Representa tati tion Reas asoning Technique SLS

slide-39
SLIDE 39

CPSC 322, Lecture 18 Slide 39

Answerin ing Query u y unde der Un Uncertai ainty

Sta tati tic B Belief Netw twork

& V Variab able le Elimi mina nation ion Dynamic mic Bayesia ian n Network rk Probab abil ility ity Theory ry Hidden n Markov

  • v Models

Email il spam m filters rs Diagno nosti stic c Sy Systems ms (e.g., ., medici cine ne) Natural al Language age Processin ssing St Student t Tracing ng in tutorin ing g Sy Systems ms Monitori

  • ring

ng (e.g credit t cards) BioInforma rmati tics cs

slide-40
SLIDE 40

CPSC 322, Lecture 29 Slide 40

Ne Next xt Cl Clas ass

Proba babi bili lity y an and d Ti Time (TextBook 6.5)

  • Work on Practice Exe

xercise se 6.C on variable elimination.

  • As

Assi sign gnment 4 4 is available on Connect. Due Sunday, , June 2 25th @ @ 11:59 pm

  • pm. Late su

submiss ssion

  • ns

s will not

  • t be

accepted, a , and late days s may not

  • t be use
  • sed. This is

due to next week being exam week, and we want to be able to release the solutions immediately .

  • Please, Fill out teaching

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