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Tutorial #9by Ma’ayan Fishelson
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Bucket Elimination Algorithm
• An algorithm for performing inference in a Bayesian network.
• Similar algorithms can be used in other domains, such as constraint satisfaction.
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Bayesian Network• X = {X1,…,Xn} is a set of random variables.
• A BN is a pair (G,P):
– G is a directed acyclic graph over nodes that
represent the random variables X.
– P = {Pi|1 ≤i ≤n}. Pi, defined on , is the
conditional probability table associated with node Xi.
Pi = P(Xi | pa(Xi))
• The BN represents a probability distribution over X.
XAi
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Bayesian Network - Example
A
B C
E
F
D
Bayesian NetworkP(A)
P(C|A)P(B|A)
P(E|B,C)
P(F|E)
P(D|A,B)
P(A,B,C,D,E,F) = P(A)P(B|A)P(C|A)P(D|A,B)P(E|B,C)P(F|E)
A
B C
E
F
D
Moralized Graph
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Some Probabilistic Inference Tasks…
• Belief Assessment: find P(Xi = xi | e).
• Most Probable Explanation (MPE):find ).,(maxarg* exPx
x
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Solution – Bucket Elimination Algorithms
Given an ordering of the variables X1,…Xn:
– Distribute P1,…,Pn into buckets B1,…,Bn.
is the highest in order in Ai.
– Process the buckets in reverse order: BnB1.
(When processing bucket Bi, multiply all the probability tables in Bi and eliminate the bucket’s variable Xi by summing over all its possible values.)
– Place the resulting function in the bucket of the highest variable (in the order) that is in its scope.
jji XBP
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Example – Belief Assessment
• Compute:
P(A=a) = Σ{A=a,B,C,D,E,F}P(A,B,C,D,E,F) = Σ{A=a,B,C,D,E,F}P(A)P(B|A)P(C|A)P(D|A,B)P(E|B,C)P(F|E)
• Suppose an order A,C,B,E,D,F. and evidence that F=1.
The distribution into buckets is as follows:
P(F|E)P(D|A,B)P(E|B,C)P(B|A)P(A) P(C|A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
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Example – Computing P(A=a)
P(D|A,B)P(E|B,C)
λF(E)P(B|A)P(A) P(C|A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
P(F|E)P(D|A,B)P(E|B,C)P(B|A)P(A) P(C|A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
• To process B6: Assign F=1, get λF(E) = P(F=1|E)
• Place λF(E) in bucket B4(E).
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Computing P(A=a) (cont. 1)
P(E|B,C)λF(E)
P(B|A)λD(A,B)P(A) P(C|A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
P(D|A,B)P(E|B,C)
λF(E)P(B|A)P(A) P(C|A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
• To process B5: λD(A,B) = ΣDP(D|A,B)
• Place λD(A,B) in bucket B3(B).
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Computing P(A=a) (cont. 2)
B3(B)
P(B|A)λD(A,B)λE(B,C)
P(A) P(C|A)
B1(A) B2(C) B4(E) B5(D) B6(F)
P(E|B,C)λF(E)
P(B|A)λD(A,B)P(A) P(C|A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
• To process B4: λE(B,C) = ΣEP(E|B,C) λF(E)
• Place λE(B,C) in bucket B3(B).
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Computing P(A=a) (cont. 3)
P(A)P(C|A)
λB(A,C)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
• To process B3: λB(A,C) = ΣBP(B|A) λD(A,B) λE(B,C) • Place λB(A,C) in bucket B2(C).
P(A) P(C|A)
B1(A) B2(C)B3(B)
B4(E) B5(D) B6(F)
P(B|A)λD(A,B)λE(B,C)
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Computing P(A=a) (cont. 4)
P(A)λC(A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
• To process B2: λC(A) = ΣCP(C|A) λB(A,C)
• Place λC(A) in bucket B1(A).
P(A)P(C|A)λB(A,C)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
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Computing P(A=a) (cont. 5)
P(A)λC(A)
B1(A) B2(C) B3(B) B4(E) B5(D) B6(F)
• To compute the belief P(A=a), we multiply P(A) λC(A).• We obtain a function of A.
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Bucket Elimination Complexity• The time and space complexity is bounded by
the size of the probability tables that are created by the algorithm.
• w*(d) is the induced width (max clique size) of the moral graph along ordering d.
))(exp(w ( * dnO
A
B C
E
Moral graph
D
B
C
D
E
AOrder B,C,D,E,A.w*(d1) =4
E
D
C
B
AOrder E,D,C,B,A.w*(d2) =2
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Constraint Satisfaction Problem (CSP)
• Input:– A set of variables: X = {X1,…,Xn}.
– A set of constraints: R = {R1,…,Rm}.
– For each constraint Ri:
• Scope of Ri:
• Ri is a relation over the variables in Ai.
• Output:– An assignment to X that is consistent with R.
.XAi
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Constraint Graph
• Nodes: {X1,…Xn}.
• Edges: kjiji AXXkEXX , s.t. ,),(
For simplicity, assume that constraints are defined on maximal cliques of the graph.
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Example of CSP: Find Possible Genotypes
• Possible haplotypes:h1=A,A1; h2 = A,A2;
h3=O,A1; h4 = O,A2;
• Possible genotypes (ordered):gij = hi/hj (father/mother)
Ind #1: g11, g13, g31
Ind #2: g44
Ind #3: g12, g21, g14, g41, g23, g32
Ind #4: g22, g24, g42
Ind #5: g34, g43
1 2
3 4
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AA1/A1
OA2/A2
AA1/A2
AA2/A2
OA1/A2
A pedigree typed at the ABO and AK1 loci:
Constraints: R13 = {<g11,g{12,14}>, <g{13,31},g{12,14,32}>},similarly for R23, R35, and R45.
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Bucket Elimination for CSP
• Given an order X1,…, Xn:
– Distribute constraints into buckets, according to the highest variable in the constraint’s scope.
– Process the buckets one by one starting from Xn:
• Join the constraints (with equal value for the eliminated variable), in the eliminated bucket and remember the value of the eliminated variable.
• Put the new constraint in the bucket of the highest variable in its scope.
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Example of CSP: Find Possible Genotypes (cont.
1)
Constraints:• RAC = {<g11,g{12,14}>,
<g{13,31},g{12,14,32}>};
• RBC = {<g44, g14>};
• RCE = {<g{14,41,23,32} g{34,43}};
• RDE = {<g{24,42} g34>}
A B
C
E
D
Constraint graph
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Find Possible Genotypes (cont. 2)
• Suppose an order: A, B, C, D, E.
• The distribution into buckets is as follows:
BABBBCBDBE
InitialRAC = {<g11,g{12,14}>,
<g{13,31},g{12,14,32}>}
RBC = {<g44, g14>}
RCE =
{<g{23,32,14,41},g{34,43}>}
RDE = {<g{24,42} g34>}
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Find Possible Genotypes (cont. 3)
BABBBCBDBE
InitialRAC = {<g11,g{12,14}>,
<g{13,31},g{12,14,32}>}
RBC = {<g44, g14>}
RCE =
{<g{23,32,14,41},g{34,43}>}
RDE = {<g{24,42}, g34>}
When processing BE:• compute RCD = {<g{23,32,14,41}, g{24,42}>}.• put RCD into bucket BD.• remember the value g34 for E.
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Find Possible Genotypes (cont. 4)
BABBBCBDBE
AfterProcessin
g BE.
RAC = {<g11,g{12,14}>,
<g{13,31},g{12,14,32}>}
RBC = {<g44, g14>}
RCD =
{<g{23,32,14,41},g{24,42}>}
When processing BD:• compute RC = {<g{23,32,14,41}}.• put RC into bucket BC.• remember the values g24,g42 for D.
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Find Possible Genotypes (cont. 5)
BABBBCBDBE
AfterProcessin
g BD.
RAC = {<g11,g{12,14}>,
<g{13,31},g{12,14,32}>}
RBC = {<g44, g14>}
RC = {<g{23,32,14,41}}.
When processing BC:• compute RAB = {<g{11,13, 31}, g44>}.• put RC into bucket BB.• remember the value g14 for C.
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Find Possible Genotypes (cont. 6)
BABBBCBDBE
AfterProcessin
g BD.
RAB = {<g{11,13, 31}, g44>}.
When processing BB:• compute RA = {<g{11,13, 31}>}.• remember the value g44 for B.
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Find Possible Genotypes (cont. 7)
By backtracking we get thepossible values:
• Ind #1 (A): g11, g13, g31
• Ind #2 (B): g44
• Ind #3 (C): g14
• Ind #4 (D): g24, g42
• Ind #5 (E): g34
1 2
3 4
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AA1/A1
OA2/A2
AA1/A2
AA2/A2
OA1/A2
1 2
3 4
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A|A or A|O or O|AA1|A1
O|OA2|A2
A|OA1|A2
A|O or O|AA2/A2
O|OA1|A2
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Bucket Elimination Algorithm
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Bucket Elimination Algorithm
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Bucket Elimination Algorithm
• More variations of the algorithm
• http://www.ics.uci.edu/~csp/R48a.pdf
