Approximation Techniques bounded inference

1
Approximation Techniques bounded inference

• COMPSCI 276
• Fall 2007

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Approximate Inference

• Metrics of evaluation
• Absolute error give egt0 and a query p P(xe),
  an estimate r has absolute error e iff p-rlte
• Relative error the ratio r/p in 1-e,1e.
• Dagum and Luby 1993 approximation up to a
  relative error is NP-hard.
• Absolute error is also NP-hard if error is less
  than .5

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Mini-buckets local inference

• Computation in a bucket is time and space
• exponential in the number of variables
  involved
• Therefore, partition functions in a bucket
• into mini-buckets on smaller number of
  variables
• (The idea is similar to i-consistency
• bound the size of recorded dependencies,
  Dechter 2003)

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Mini-bucket approximation MPE task
Split a bucket into mini-buckets gtbound
complexity
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Mini-Bucket Elimination
Mini-buckets
minBS
minBS
bucket B
F(b,d)
F(b,e)
F(a,b)
F(b,c)
A
hB(a,c)
F(c,e)
F(a,c)
bucket C
B
C
F(a,d)
hB(d,e)
bucket D
E
D
hC(e,a)
e 0
hD(e,a)
bucket E
hE(a)
bucket A
L lower bound
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Semantics of Mini-Bucket Splitting a Node
Variables in different buckets are renamed and
duplicated (Kask et. al., 2001), (Geffner et.
al., 2007), (Choi, Chavira, Darwiche , 2007)
After SplittingNetwork N'
Before SplittingNetwork N
U
U
Û
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Approx-mpe(i)

• Input i max number of variables allowed in a
  mini-bucket
• Output lower bound (P of a sub-optimal
  solution), upper bound

Example approx-mpe(3) versus elim-mpe
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MBE(i,m), (MBE(i) (also, approx-mpe)

• Input Belief network (P1,Pn)
• Output upper and lower bounds
• Initialize (put functions in buckets)
• Process each bucket from pn to 1
• Create (i,m)-mini-buckets
• Process each mini-bucket
• (For mpe) assign values in ordering d
• Return mpe-tuple, upper and lower bounds

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Properties of approx-mpe(i)

• Complexity O(exp(2i)) time and O(exp(i))
  space.
• Accuracy determined by upper/lower (U/L) bound.
• As i increases, both accuracy and complexity
  increase.
• Possible use of mini-bucket approximations
• As anytime algorithms (Dechter and Rish, 1997)
• As heuristics in best-first search (Kask and
  Dechter, 1999)
• Other tasks similar mini-bucket approximations
  for belief updating, MAP and MEU (Dechter and
  Rish, 1997)

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Anytime Approximation
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Bounded Inference for belief updating

• Idea mini-bucket is the same
• So we can apply a sum in each mini-bucket, or
  better, one sum and the rest max, or min (for
  lower-bound)
• Approx-bel-max(i,m) generating upper and
  lower-bound on beliefs approximates elim-bel
• Approx-map(i,m) max buckets will be maximized,
  sum buckets will be sum-max. Approximates
  elim-map.

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Empirical Evaluation(Dechter and Rish, 1997
Rish thesis, 1999)

• Randomly generated networks
• Uniform random probabilities
• Random noisy-OR
• CPCS networks
• Probabilistic decoding
• Comparing approx-mpe and anytime-mpe
• versus elim-mpe

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Causal Independence

• Event X has two possible causes A,B. It is hard
  to elicit P(XA,B) but it is easy to determine
  P(XA) and P(XB).
• Example several diseases causes a symptom.
• Effect of A on X is independent from the effect
  of B on X
• Causal Independence, using canonical models
• Noisy-O, Noisy AND, noisy-max

A
B
X
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Binary OR
A
B
X
A
B
P(X0A,B)
P(X1A,B)
0
0
1
0
0
1
0
1
1
0
0
1
1
1
0
1
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Noisy-OR

• noise is associated with each edge
• described by noise parameter ? ? 0,1
• Let q b0.2, qa 0.1
• P(x0a,b) (1-?a) (1-?b)
• P(x1a,b)1-(1-?a) (1-?b)

A
B
?a
?b
X
A
B
P(X0A,B)
P(X1A,B)
0
0
1
0
0
1
0.1
0.9
qiP(X0A_i1,else 0)
1
0
0.2
0.8
1
1
0.02
0.98
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Closed Form Bel(X) - 1
Given noisy-or CPT P(xu) noise parameters
?i Tu i Ui 1 Define qi 1 - ?I Then
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Closed Form Bel(X) - 2
Using Iterative Belief Propagation
Set piix pix (uk1). Then we can show that
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Methodology for Empirical Evakuation (for mpe)

• U/L accuracy
• Better (U/mpe) or mpe/L
• Benchmarks Random networks
• Given n,e,v generate a random DAG
• For xi and parents generate table from uniform
  0,1, or noisy-or
• Create k instances. For each generate random
  evidence, likely evidence
• Measure averages

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Random networks

• Uniform random 60 nodes, 90 edges (200
  instances)
• In 80 of cases, 10-100 times speed-up while
  U/Llt2
• Noisy-OR even better results
• Exact elim-mpe was infeasible appprox-mpe took
  0.1 to 80 sec.

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CPCS networks medical diagnosis(noisy-OR model)
Test case no evidence
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The effect of evidence
More likely evidencegthigher MPE gt higher
accuracy (why?)
Likely evidence versus random (unlikely) evidence
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Probabilistic decoding
Error-correcting linear block code
State-of-the-art
approximate algorithm iterative belief
propagation (IBP) (Pearls poly-tree algorithm
applied to loopy networks)
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Iterative Belief Proapagation

• Belief propagation is exact for poly-trees
• IBP - applying BP iteratively to cyclic networks
• No guarantees for convergence
• Works well for many coding networks

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approx-mpe vs. IBP
Bit error rate (BER) as a function of noise
(sigma)
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Mini-buckets summary

• Mini-buckets local inference approximation
• Idea bound size of recorded functions
• Approx-mpe(i) - mini-bucket algorithm for MPE
• Better results for noisy-OR than for random
  problems
• Accuracy increases with decreasing noise in
  coding
• Accuracy increases for likely evidence
• Sparser graphs -gt higher accuracy
• Coding networks approx-mpe outperfroms IBP on
  low-induced width codes

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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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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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Mini-Clustering for belief updating

• 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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Idea of Mini-Clustering
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Mini-Clustering - MC
Mini-Clustering, i3
A B C p(a), p(ba), p(ca,b)
Cluster Tree Elimination
1
BC
B C D F p(db), p(fc,d)
B C D F p(db), h(1,2)(b,c), p(fc,d)
2
2
BF
sep(2,3) B,F elim(2,3) C,D
B E F p(eb,f)
3
EF
E F G p(ge,f)
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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Mini-Clustering - example
ABC
1
BC
BCDF
2
BF
BEF
3
EF
EFG
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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Lower bounds and mean approximations

• We can replace max operator by
• min gt lower bound on the joint
• mean gt approximation of the joint

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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

• 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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Noisy-OR networks - Absolute error
evidence10
evidence20
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Grid 15x15 - 10 evidence
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CPCS422 - Absolute error
evidence0
evidence10
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Coding networks - Bit Error Rate
sigma0.22
sigma.51
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Mini-Clustering summary

• 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

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What is IJGP?

• IJGP is an approximate algorithm for belief
  updating in Bayesian networks
• IJGP is a version of join-tree clustering which
  is both anytime and iterative
• IJGP applies message passing along a join-graph,
  rather than a join-tree
• Empirical evaluation shows that IJGP is almost
  always superior to other approximate schemes
  (IBP, MC)

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Iterative Belief Propagation - IBP
One step update BEL(U1)
U1
U2
U3

• Belief propagation is exact for poly-trees
• IBP - applying BP iteratively to cyclic networks
• No guarantees for convergence
• Works well for many coding networks

X2
X1
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IJGP - Motivation

• IBP is applied to a loopy network iteratively
• not an anytime algorithm
• when it converges, it converges very fast
• MC applies bounded inference along a tree
  decomposition
• MC is an anytime algorithm controlled by i-bound
• MC converges in two passes up and down the tree
• IJGP combines
• the iterative feature of IBP
• the anytime feature of MC

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IJGP - The basic idea

• Apply Cluster Tree Elimination to any join-graph
• We commit to graphs that are minimal I-maps
• Avoid cycles as long as I-mapness is not violated
• Result use minimal arc-labeled join-graphs

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IJGP - Example
A
B
C
A
C
A
ABC
C
A
AB
BC
C
D
E
ABDE
BCE
BE
C
DE
CE
F
CDEF
G
H
H
FGH
H
F
F
FG
GH
H
GI
I
J
FGI
GHIJ
a) Belief network
a) The graph IBP works on
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Arc-minimal join-graph
A
C
A
A
ABC
C
A
ABC
C
A
AB
BC
C
AB
BC
ABDE
BCE
ABDE
BCE
BE
C
C
DE
CE
DE
CE
CDEF
CDEF
H
H
FGH
H
FGH
H
F
F
F
FG
GH
H
FG
GH
GI
GI
FGI
GHIJ
FGI
GHIJ
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Minimal arc-labeled join-graph
A
A
A
ABC
C
A
ABC
C
AB
BC
AB
BC
ABDE
BCE
ABDE
BCE
C
C
DE
CE
DE
CE
CDEF
CDEF
H
H
FGH
H
FGH
H
F
F
FG
GH
F
GH
GI
GI
FGI
GHIJ
FGI
GHIJ
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Join-graph decompositions
A
A
ABC
C
AB
BC
BC
BC
ABDE
BCE
ABCDE
BCE
ABCDE
BCE
C
DE
CE
CDE
CE
DE
CE
CDEF
CDEF
CDEF
H
FGH
H
FGH
FGH
F
F
F
F
GH
F
GH
F
GH
GI
GI
GI
FGI
GHIJ
FGI
GHIJ
FGI
GHIJ
a) Minimal arc-labeled join graph
b) Join-graph obtained by collapsing nodes of
graph a)
c) Minimal arc-labeled join graph
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Tree decomposition
BC
ABCDE
BCE
ABCDE
DE
CE
CDE
CDEF
CDEF
FGH
F
F
F
GH
GHI
GI
FGI
GHIJ
FGHI
GHIJ
a) Minimal arc-labeled join graph
a) Tree decomposition
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Join-graphs
more accuracy
less complexity
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Message propagation
BC
ABCDE
BCE
ABCDE p(a), p(c), p(bac), p(dabe),p(eb,c)
h(3,1)(bc)
h(3,1)(bc)
CDE
BCD
1
3
CE
BC
CDEF
FGH
h(1,2)
CDE
CE
F
F
GH
CDEF
2
GI
FGI
GHIJ
Minimal arc-labeled sep(1,2)D,E
elim(1,2)A,B,C
Non-minimal arc-labeled sep(1,2)C,D,E
elim(1,2)A,B
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Bounded decompositions

• We want arc-labeled decompositions such that
• the cluster size (internal width) is bounded by i
  (the accuracy parameter)
• the width of the decomposition as a graph
  (external width) is as small as possible
• Possible approaches to build decompositions
• partition-based algorithms - inspired by the
  mini-bucket decomposition
• grouping-based algorithms

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Partition-based algorithms
GFE
P(GF,E)
EF
EBF
P(EB,F)
P(FC,D)
BF
F
FCD
BF
CD
CDB
P(DB)
CB
B
CAB
P(CA,B)
BA
BA
P(BA)
A
A
P(A)
a) schematic mini-bucket(i), i3 b) arc-labeled
join-graph decomposition
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IJGP properties

• IJGP(i) applies BP to min arc-labeled
  join-graph, whose cluster size is bounded by i
• On join-trees IJGP finds exact beliefs
• IJGP is a Generalized Belief Propagation
  algorithm (Yedidia, Freeman, Weiss 2001)
• Complexity of one iteration
• time O(deg(nN) d i1)
• space O(Nd?)

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Empirical evaluation

• Measures
• Absolute error
• Relative error
• Kulbach-Leibler (KL) distance
• Bit Error Rate
• Time

• Algorithms
• Exact
• IBP
• MC
• IJGP

• Networks (all variables are binary)
• Random networks
• Grid networks (MxM)
• CPCS 54, 360, 422
• Coding networks

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Random networks - KL at convergence
evidence0
evidence5
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Random networks - KL vs. iterations
evidence0
evidence5
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Random networks - Time
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Coding networks - BER
sigma.22
sigma.32
sigma.51
sigma.65
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Coding networks - Time
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IJGP summary

• IJGP borrows the iterative feature from IBP and
  the anytime virtues of bounded inference from MC
• Empirical evaluation showed the potential of
  IJGP, which improves with iteration and most of
  the time with i-bound, and scales up to large
  networks
• IJGP is almost always superior, often by a high
  margin, to IBP and MC
• Based on all our experiments, we think that IJGP
  provides a practical breakthrough to the task of
  belief updating

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Heuristic search

• Mini-buckets record upper-bound heuristics
• The evaluation function over
• Best-first expand a node with maximal evaluation
  function
• Branch and Bound prune if f gt upper bound
• Properties
• an exact algorithm
• Better heuristics lead to more prunning

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Heuristic Function
Given a cost function
P(a,b,c,d,e) P(a) P(ba) P(ca) P(eb,c)
P(db,a)
Define an evaluation function over a partial
assignment as the probability of its best
extension
0
D
0
B
E
0
D
1
A
B
1
D
E
1
D
f(a,e,d) maxb,c P(a,b,c,d,e) P(a)
maxb,c P)ba) P(ca) P(eb,c) P(da,b)
g(a,e,d) H(a,e,d)
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Heuristic Function
H(a,e,d) maxb,c P(ba) P(ca) P(eb,c)
P(da,b) maxc P(ca) maxb P(eb,c)
P(ba) P(da,b) maxc P(ca) maxb
P(eb,c) maxb P(ba) P(da,b)
H(a,e,d) f(a,e,d) g(a,e,d) H(a,e,d) ³
f(a,e,d) The heuristic function H is compiled
during the preprocessing stage of the
Mini-Bucket algorithm.
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Heuristic Function
The evaluation function f(xp) can be computed
using function recorded by the Mini-Bucket scheme
and can be used to estimate the probability of
the best extension of partial assignment xpx1,
, xp,
f(xp)g(xp) H(xp )
For example,
maxB P(eb,c) P(da,b)
P(ba) maxC P(ca) hB(e,c) maxD

hB(d,a) maxE hC(e,a) maxA P(a)
hE(a) hD (a)
H(a,e,d) hB(d,a) hC (e,a)
g(a,e,d) P(a)
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Properties

• Heuristic is monotone
• Heuristic is admissible
• Heuristic is computed in linear time
• IMPORTANT
• Mini-buckets generate heuristics of varying
  strength using control parameter bound I
• Higher bound -gt more preprocessing -gt
• stronger heuristics -gt less search
• Allows controlled trade-off between preprocessing
  and search

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Empirical Evaluation of mini-bucket heuristics