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Exact Inference Algorithms Exact Inference Algorithms for Probabilistic Reasoning; for Probabilistic Reasoning;

COMPSCI 276Fall 2007

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Probabilistic Inference Tasks

X/A

a

*k

*1 e),xP(maxarg)a,...,(a

evidence)|xP(X)BEL(X iii

Belief updating:

Finding most probable explanation (MPE)

Finding maximum a-posteriory hypothesis

Finding maximum-expected-utility (MEU) decision

e),xP(maxarg*xx

)xU(e),xP(maxarg)d,...,(d X/D

d

*k

*1

variableshypothesis: XA

function utilityx variablesdecision

: )( :

UXD

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Example with a chain

A B C D

P(D)=? P(D|A=a)=?P(A|D=d)=?

O(4k^2) instead of O(k^4), k is the domain size

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Example of product-sum in a bucket

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Bucket elimination Algorithm elim-bel (Dechter 1996)

b

Elimination operator

P(a|e=0)

W*=4”induced width” (max clique size)

bucket B:

P(a)

P(c|a)

P(b|a) P(d|b,a) P(e|b,c)

bucket C:

bucket D:

bucket E:

bucket A:

e=0

B

C

D

E

A

e)(a,hD

(a)hE

e)c,d,(a,hB

e)d,(a,hC

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E

D

C

B

A

B

C

D

E

A

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Complexity of elimination))((exp ( * dwnO

ddw ordering along graph moral of widthinduced the)(*

The effect of the ordering:

4)( 1* dw 2)( 2

* dw“Moral” graph

A

D E

CB

B

C

D

E

A

E

D

C

B

A

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Finding small induced-width NP-complete A tree has induced-width of ? Greedy algorithms:

Min width Min induced-width Max-cardinality Fill-in (thought as the best) See anytime min-width (Gogate and

Dechter)

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Different Induced graphs

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The impact of observations

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“Moral” graph

A

D E

CB

Theorem: elim-bel is exponential in the adjusted induced-width w*(e,d)

21Use the ancestral graph only

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BTE in action