Two Approximate Algorithms for Belief Updating

1
Two Approximate Algorithms for Belief Updating

• Mini-Clustering - MC
• Robert Mateescu, Rina Dechter, Kalev Kask. "Tree
  Approximation for Belief Updating", AAAI-2002
• Iterative Join-Graph Propagation - IJGP
• Rina Dechter, Kalev Kask and Robert Mateescu.
  "Iterative Join-Graph Propagation, UAI 2002

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What is Mini-Clustering?

• Mini-Clustering (MC) is an approximate algorithm
  for belief updating in Bayesian networks
• MC is an anytime version of join-tree clustering
• MC applies message passing along a cluster tree
• The complexity of MC is controlled by a
  user-adjustable parameter, the i-bound
• Empirical evaluation shows that MC is a very
  effective algorithm, in many cases superior to
  other approximate schemes (IBP, Gibbs Sampling)

3
Motivation

• Probabilistic reasoning using belief networks is
  known to be NP-hard
• Nevertheless, approximate inference can be a
  powerful tool for decision making under
  uncertainty
• We propose an anytime version of Cluster Tree
  Elimination

4
Outline

• Preliminaries
• Belief networks
• Tree decompositions
• Tree Clustering algorithm
• Mini-Clustering algorithm
• Experimental results

5
Belief networks

• The belief updating problem is the task of
  computing the posterior probability P(Ye) of
  query nodes Y ? X given evidence e.We focus on
  the basic case where Y is a single variable Xi

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Tree decompositions
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Tree decompositions
A B C p(a), p(ba), p(ca,b)
BC
B C D F p(db), p(fc,d)
BF
B E F p(eb,f)
A
B
EF
E F G p(ge,f)
E
C
D
F
Belief network
Tree decomposition
G
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Example Join-tree
A B C p(a), p(ba), p(ca,b)
BC
B C D F p(db), p(fc,d)
BF
B E F p(eb,f)
EF
E F G p(ge,f)
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Cluster Tree Elimination

• Cluster Tree Elimination (CTE) is an exact
  algorithm that works by passing messages along a
  tree decomposition
• Basic idea
• Each node sends only one message to each of its
  neighbors
• Node u sends a message to its neighbor v only
  when u received messages from all its other
  neighbors
• Previous work on tree clustering
• Lauritzen, Spiegelhalter - 88 (probabilities)
• Jensen, Lauritzen, Olesen - 90 (probabilities)
• Shenoy, Shafer - 90, Shenoy - 97 (general)
• Dechter, Pearl - 89 (constraints)
• Gottlob, Leone, Scarello - 00 (constraints)

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Cluster Tree Elimination

• Cluster Tree Elimination (CTE) is an exact
  algorithm
• It works by passing messages along a tree
  decomposition
• Basic idea
• Each node sends only one message to each of its
  neighbors
• Node u sends a message to its neighbor v only
  when u received messages from all its other
  neighbors

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Cluster Tree Elimination

• Previous work on tree clustering
• Lauritzen, Spiegelhalter - 88 (probabilities)
• Jensen, Lauritzen, Olesen - 90 (probabilities)
• Shenoy, Shafer - 90, Shenoy - 97 (general)
• Dechter, Pearl - 89 (constraints)
• Gottlob, Leone, Scarello - 00 (constraints)

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Belief Propagation
x1
h(u,v)
v
u
x2
xn
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Belief Propagation
x1
h(u,v)
v
u
x2
xn
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Cluster Tree Elimination - example
ABC
1
BC
BCDF
2
BF
BEF
3
EF
EFG
4
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Cluster Tree Elimination - the messages
A B C p(a), p(ba), p(ca,b)
1
BC
B C D F p(db), p(fc,d) h(1,2)(b,c)
2
sep(2,3)B,F elim(2,3)C,D
BF
B E F p(eb,f), h(2,3)(b,f)
3
EF
E F G p(ge,f)
4
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Cluster Tree Elimination - properties

• Correctness and completeness Algorithm CTE is
  correct, i.e. it computes the exact joint
  probability of a single variable and the
  evidence.
• Time complexity O ( deg ? (nN) ? d w1 )
• Space complexity O ( N ? d sep)
• where deg the maximum degree of a node
• n number of variables ( number of CPTs)
• N number of nodes in the tree decomposition
• d the maximum domain size of a variable
• w the induced width
• sep the separator size

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Mini-Clustering - motivation

• Time and space complexity of Cluster Tree
  Elimination depend on the induced width w of the
  problem
• When the induced width w is big, CTE algorithm
  becomes infeasible

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Mini-Clustering - the basic idea

• Try to reduce the size of the cluster (the
  exponent) partition each cluster into
  mini-clusters with less variables
• Accuracy parameter i maximum number of
  variables in a mini-cluster
• The idea was explored for variable elimination
  (Mini-Bucket)

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Mini-Clustering

• Motivation
• Time and space complexity of Cluster Tree
  Elimination depend on the induced width w of the
  problem
• When the induced width w is big, CTE algorithm
  becomes infeasible
• The basic idea
• Try to reduce the size of the cluster (the
  exponent) partition each cluster into
  mini-clusters with less variables
• Accuracy parameter i maximum number of
  variables in a mini-cluster
• The idea was explored for variable elimination
  (Mini-Bucket)

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Mini-Clustering

• Suppose cluster(u) is partitioned into p
  mini-clusters mc(1),,mc(p), each containing at
  most i variables
• TC computes the exact message
• We want to process each ?f?mc(k) f separately

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Mini-Clustering

• Approximate each ?f?mc(k) f , k2,,p and take it
  outside the summation
• How to process the mini-clusters to obtain
  approximations or bounds
• Process all mini-clusters by summation - this
  gives an upper bound on the joint probability
• A tighter upper bound process one mini-cluster
  by summation and the others by maximization
• Can also use mean operator (average) - this gives
  an approximation of the joint probability

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Idea of Mini-Clustering
Split a cluster into mini-clusters gtbound
complexity
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Mini-Clustering - example
ABC
1
BC
BCDF
2
BF
BEF
3
EF
EFG
4
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Mini-Clustering - the messages, i3
A B C p(a), p(ba), p(ca,b)
1
BC
B C D p(db), h(1,2)(b,c) C D F p(fc,d)
2
sep(2,3)B,F elim(2,3)C,D
BF
B E F p(eb,f), h1(2,3)(b), h2(2,3)(f)
3
EF
E F G p(ge,f)
4
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Cluster Tree Elimination vs. Mini-Clustering
ABC
ABC
1
1
BC
BC
BCDF
BCDF
2
2
BF
BF
BEF
BEF
3
3
EF
EF
EFG
EFG
4
4
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Mini-Clustering

• Correctness and completeness Algorithm MC(i)
  computes a bound (or an approximation) on the
  joint probability P(Xi,e) of each variable and
  each of its values.
• Time space complexity O(n ? hw ? d i)
• where hw maxu f f ? ?(u) ? ?

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Normalization

• Algorithms for the belief updating problem
  compute, in general, the joint probability
• Computing the conditional probability
• is easy to do if exact algorithms can be applied
• becomes an important issue for approximate
  algorithms

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Normalization

• MC can compute an (upper) bound on
  the joint P(Xi,e)
• Deriving a bound on the conditional P(Xie) is
  not easy when the exact P(e) is not available
• If a lower bound would be available, we
  could useas an upper bound on the posterior
• In our experiments we normalized the results and
  regarded them as approximations of the posterior
  P(Xie)

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Experimental results

• We tested MC with max and mean operators
• Algorithms
• Exact
• IBP
• Gibbs sampling (GS)
• MC with normalization (approximate)
• Networks (all variables are binary)
• Coding networks
• CPCS 54, 360, 422
• Grid networks (MxM)
• Random noisy-OR networks
• Random networks

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Experimental results

• Measures
• Normalized Hamming Distance
• pick most likely value (for exact and for
  approximate)
• take ratio between number of disagreements and
  total number of variables
• average over problems
• BER (Bit Error Rate) - for coding networks
• Absolute error
• difference between exact and the approximate,
  averaged over all values, all variables, all
  problems
• Relative error
• difference between exact and the approximate,
  divided by the exact, averaged over all values,
  all variables, all problems
• Time

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Experimental results
We tested MC with max and mean operators

• Algorithms
• Exact
• IBP
• Gibbs sampling (GS)
• MC with normalization (approximate)
• Networks (all variables are binary)
• Coding networks
• CPCS 54, 360, 422
• Grid networks (MxM)
• Random noisy-OR networks
• Random networks

• Measures
• Normalized Hamming Distance (NHD)
• BER (Bit Error Rate)
• Absolute error
• Relative error
• Time

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Random networks - Absolute error
evidence0
evidence10
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Coding networks - Bit Error Rate
sigma0.22
sigma.51
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Noisy-OR networks - Absolute error
evidence10
evidence20
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CPCS422 - Absolute error
evidence0
evidence10
36
Grid 15x15 - 0 evidence
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Grid 15x15 - 10 evidence
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Grid 15x15 - 20 evidence
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Coding Networks 1
N100, P3, w7
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Coding Networks 2
N100, P4, w11
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CPCS54 - w15
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Noisy-OR Networks 1
N50, P2, w10
43
Noisy-OR Networks 2
N50, P3, w16
44
Random Networks 1
N50, P2, w10
45
Random Networks 2
N50, P3, w16
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Conclusion

• MC extends the partition based approximation from
  mini-buckets to general tree decompositions for
  the problem of belief updating
• Empirical evaluation demonstrates its
  effectiveness and superiority (for certain types
  of problems, with respect to the measures
  considered) relative to other existing algorithms