Reasoning Under Uncertainty: Variable elimination

1
Reasoning Under Uncertainty Variable
elimination Computer Science cpsc322, Lecture
30 (Textbook Chpt 10.4) March, 28, 2008
2
Lecture Overview

• Recap Intro Variable Elimination
• Variable Elimination
• Simplifications
• Example
• Independence
• Where are we?

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Last Lecture Summary
Computing P(Z Y1v1 ? ? Yjvj ) can be
reduced to computing P(Z ? Y1v1 ? ? Yjvj )
The computation of P(Z ? Y1v1 ? ? Yjvj )
can be expressed in terms of factors and the
three basic operations between factors

• Bnet inference as variable elimination
  (preliminary)
• Construct a factor for each conditional
  probability of the Bnet.
• In each factor assign the observed variables to
  their observed values.
• Multiply the factors and for each of the other
  variables Zi ? Z1, , Zk , sum out Zi

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Variable Elimination Intro

• Using the chain rule and the definition of a
  belief network, we can write P(X1, , Xn) as
• Thus

Consider each CPT as a factor Inference in
belief networks thus reduces to computing the
sums of products.
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Lecture Overview

• Recap Intro Variable Elimination
• Variable Elimination
• Simplifications
• Example
• Independence
• Where are we?

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How to simplify the Computation?

• Assume we have turned the CPTs into factors and
  performed the assignments

• Lets focus on the basic case.

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How to simplify basic case

• Lets focus on the basic case.

• How can we compute efficiently?

Factor out those terms that don't involve Z1 !
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General case Summing out variables efficiently

• Now to sum out a variable Z2 from a product f1
  fi f of factors, again partition the factors
  into
• those that don't contain Z2
• those that contain Z2

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Analogy with Computing sums of products

• This simplification is similar to what you can do
  in basic algebra with and x
• 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.
• How can this expression be evaluated more
  efficiently?

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Variable elimination algorithm Summary

• To compute P(Q Y1v1 , ,Yjvj )
• Construct a factor for each conditional
  probability.
• Set the observed variables to their observed
  values.
• For each of the other variables Zi ? Z1, , Zk
  , sum out Zi
• Multiply the remaining factors.
• Normalize by dividing the resulting factor f(Q)
  by ?Q f(Q) .

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Lecture Overview

• Recap Intro Variable Elimination
• Variable Elimination
• Simplifications
• Example
• Independence
• Where are we?

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Variable elimination example

• Compute P(G Hh1 ).
• P(G,H) ?A,B,C,D,E,F,I P(A,B,C,D,E,F,G,H,I)

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Variable elimination example

• Compute P(G Hh1 ).
• P(G,H) ?A,B,C,D,E,F,I P(A,B,C,D,E,F,G,H,I)
• Chain Rule
• P(G,H) ?A,B,C,D,E,F,I P(A)P(BA)P(C)P(DB,C)P(E
  C)P(FD)P(GF,E)P(HG)P(IG)

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Variable elimination example

• Compute P(G Hh1 ).
• P(G,H) ?A,B,C,D,E,F,I P(A)P(BA)P(C)P(DB,C)P(E
  C)P(FD)P(GF,E)P(HG)P(IG)
• 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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Variable elimination example

• Compute P(G Hh1 ).
• 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,Hh1) ?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)

• 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)

• f9(G)

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Variable elimination example

• Compute P(G Hh1 ).
• 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,Hh1) 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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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state
• P(G,Hh1) 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,Hh1) 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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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state
• P(G,Hh1) 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,Hh1) 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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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state
• P(G,Hh1) 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,Hh1) 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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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state P(G,Hh1) f9(G) ?F ?D f5(F, D)
  ?B f10(B) f13(B,D,F,G) ?I f8(I,G)
• Eliminate I
• P(G,Hh1) 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)

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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state P(G,Hh1) f9(G) f14(G) ?F ?D
  f5(F, D) ?B f10(B) f13(B,D,F,G)
• Eliminate B
• P(G,Hh1) 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)

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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state P(G,Hh1) f9(G) f14(G) ?F ?D
  f5(F, D) f15(D,F,G)
• Eliminate D
• P(G,Hh1) 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)

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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state P(G,Hh1) f9(G) f14(G) ?F
  f16(F, G)
• Eliminate F
• P(G,Hh1) 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)

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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state P(G,Hh1) f9(G) f14(G) f17(G)
• Multiply remaining factors
• P(G,Hh1) 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)

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Variable elimination example

• Compute P(G Hh1 ). Elimination ordering A, C,
  E, I, B, D, F.
• Previous state
• P(G,Hh1) f18(G)
• Normalize
• P(G Hh1) f18(G) / ?g ? dom(G) 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)

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Lecture Overview

• Recap Intro Variable Elimination
• Variable Elimination
• Simplifications
• Example
• Independence
• Where are we?

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Variable elimination and conditional independence

• Variable Elimination looks incredibly painful for
  large graphs?
• Did we use 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

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VE and conditional independence Example

• All the variables from which the query is
  conditional independent given the observations
  can be pruned from the Bnet

• e.g., P(G Hh1, F f1 , Cc1).

• One last trick (no intuitive justification?) we
  can repeatedly eliminate all leaf nodes that are
  neither observed nor queried, until we reach a
  fixed point

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Lecture Overview

• Recap Intro Variable Elimination
• Variable Elimination
• Simplifications
• Example
• Independence
• Where are we?

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Course Big picture
Environment
Stochastic
Deterministic
Search
Single Action
Constraint Satisfaction (CSPs)
Decision
Logics
Search
Sequence of Actions
Constraint Satisfaction (CSPs)
Planning
31
Assignment 4

• It was posted a week ago - due Wed April 9
• Now you can work on all the questions except for
  the last one (on decision networks).
• If you have not done it yet, start working on it
  ASAP!