Exact Reasoning: conditioning search and hybrids

1
Exact Reasoningconditioning search and hybrids

• COMPSCI 276
• Fall 2007

2
Probabilistic Inference Tasks

• Belief updating
• Finding most probable explanation (MPE)
• Finding maximum a-posteriory hypothesis
• Finding maximum-expected-utility (MEU) decision
  

3
Belief Updating
Smoking
lung Cancer
Bronchitis
X-ray
Dyspnoea
P (lung canceryes smokingno, dyspnoeayes )
?
4
Conditioning generates the probability tree
Complexity of conditioning exponential time,
linear space
5
ConditioningElimination
6
Loop-cutset decomposition

• You condition until you get a polytree

A0
A1
A1
A1
A1
C
B
B
F
P(BF0) P(B, A0F0)P(B,A1F0)
Loop-cutset method is time exp in loop-cutset
size And linear space
7
Super-bucket elimination(Dechter and El Fattah,
1996)

• Eliminating several variables at once
• Conditioning is done only in super-buckets

8
The idea of super-buckets
Larger super-buckets (cliques) gtmore time but
less space

• Complexity
• Time exponential in clique (super-bucket) size
• Space exponential in separator size

9
OR search space
A
F
B
C
Ordering A B E C D F
D
E
10
AND/OR search space
A
F
B
C
D
E
Primal graph
DFS tree
11
AND/OR vs. OR
AND
0
1
AND/OR
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
12
OR vs AND/OR
OR
A
AND
0
1
AND/OR
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
OR
1
1
1
0
1
13
AND/OR vs. OR
AND
0
1
AND/OR
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
AND/OR size exp(4), OR size exp(6)
OR
14
AND/OR vs. OR
No-goods (A1,B1) (B0,C0)
AND
0
1
AND/OR
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
15
AND/OR vs. OR
(A1,B1) (B0,C0)
OR
A
AND
0
1
AND/OR
OR
B
B
AND
0
1
0
OR
E
C
E
C
E
C
AND
1
0
1
0
1
0
1
1
0
1
OR
D
F
D
F
D
F
D
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
A
A
A
0
1
0
1
0
1
OR
B
B
B
0
1
0
0
1
0
0
1
0
E
E
E
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
C
C
C
1
1
0
1
0
1
1
1
1
1
0
1
0
1
1
1
1
1
0
1
0
1
1
1
D
D
D
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
F
F
F
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
16
OR space vs. AND/OR space
17
AND/OR search tree for graphical models

• The AND/OR search tree of R relative to a
  spanning-tree, T, has
• Alternating levels of OR nodes (variables) and
  AND nodes (values)
• Successor function
• The successors of OR nodes X are all its
  consistent values along its path
• The successors of AND ltX,vgt are all X child
  variables in T
• A solution is a consistent subtree
• Task compute the value of the root node

18
From DFS trees to pseudo-trees (Freuder 85,
Bayardo 95)
1
4
6
1
2
3
2
7
5
3
(a) Graph
4
7
1
1
2
7
5
3
5
3
5
2
7
4
6
6
4
6
(b) DFS tree depth3
(c) pseudo- tree depth2
(d) Chain depth6
19
From DFS trees to Pseudo-trees
OR
1
AND
a
b
c
OR
2
7
AND
a
b
c
a
b
c
OR
DFS tree depth 3
3
3
3
5
5
5
AND
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
OR
4
4
4
4
4
4
4
4
4
6
6
6
6
6
6
6
6
6
AND
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
b
a
c
pseudo- tree depth 2
20
Finding min-depth backbone trees

• Finding min depth DFS, or pseudo tree is
  NP-complete, but
• Given a tree-decomposition whose tree-width is
  w, there exists a pseudo tree T of G whose
  depth, satisfies (Bayardo and Mirankar, 1996,
  bodlaender and Gilbert, 91)
• m lt w log n,

21
Generating pseudo-trees from Bucket trees
A
A
(A)
B
B
(AB)
C
C
(AC) (BC)
E
E
(AE)
(AE) (BE)
D
D
(BD) (DE)
F
F
(AF) (EF)
Bucket-tree based on d
d A B C E D F
Induced graph
Bucket-tree
Bucket-tree used as pseudo-tree
AND/OR search tree
22
Pseudo Trees vs. DFS Trees
N number of nodes, P number of parents.
MIN-FILL ordering. 100 instances.
23
AND/OR Search-tree properties (k domain size,
m pseudo-tree depth. n number of variables)

• Theorem Any AND/OR search tree based on a
  pseudo-tree is sound and complete (expresses all
  and only solutions)
• Theorem Size of AND/OR search tree is O(n km)
• Size of OR search tree is O(kn)
• Theorem Size of AND/OR search tree can be
  bounded by O(exp(w log n))
• Related to (Freuder 85 Dechter 90, Bayardo et.
  al. 96, Darwiche 1999, Bacchus 2003)
• When the pseudo-tree is a chain we get an OR
  space

24
Tasks and value of nodes

• V( n) is the value of the tree T(n) for the task
• Counting v(n) is number of solutions in T(n)
• Consistency v(n) is 0 if T(n) inconsistent, 1
  othewise.
• Optimization v(n) is the optimal solution in
  T(n)
• Belief updating v(n), probability of evidence in
  T(n).
• Partition function v(n) is the total
  probability in T(n).
• Goal compute the value of the root node
  recursively using dfs search of the AND/OR tree.
• Theorem Complexity of AO dfs search is
• Space O(n)
• Time O(n km)
• Time O(exp(w log n))

25
DFS algorithm (CSP example)
A
0
B
4
0
OR node Marginalization operator (summation)
E
C
2
2
AND node Combination operator (product)
0
0
1
0
1
2
1
1
D
F
D
F
2
1
2
0
0
1
0
1
0
1
0
1
1
1
1
0
1
1
0
0
Value of node number of solutions below it
26
Belief-updating on example
27
A
PA(0)
PA(1)
0
1
B
B
PBA(00)
PBA(10)
PBA(01)
PBA(11)
0
1
0
1
E
D
E
D
E
D
E
D
PEAB(10,0)
PEAB(00,1)
PEAB(01,0)
PEAB(11,0)
PEAB(01,1)
PEAB(11,1)
PEAB(00,0)
PEAB(10,1)
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
C
C
C
C
C
C
C
C
PDBC(00,0) PCA(00)
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
28
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29
(No Transcript)
30
AND/OR Tree DFS Algorithm (Belief Updating)
Result P(D1,E0)
Evidence E0
OR
.24408
A
.6
.4
AND
.3028
.1559
0
1
.3028
OR
.1559
B
B
.1
.9
.4
.6
AND
.352
.27
.623
.104
0
1
0
1
OR
.4
.5
.7
.2
.88
.54
.89
.52
E
C
E
C
E
C
E
C
.4
.5
.7
.2
.2
.8
.2
.8
.1
.9
.1
.9
AND
.8
.9
.7
.5
.8
.9
.7
.5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
.8
.9
.7
.5
.8
.9
.7
.5
D
D
D
D
D
D
D
D
.7
.8
.9
.5
.7
.8
.9
.5
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR node Marginalization operator (summation)
AND node Combination operator (product)
Value of node updated belief for sub-problem
below
Evidence D1
31
Complexity of AND/OR Tree Search
k domain size m depth of pseudo-tree n
number of variables w treewidth
32
From Search Trees to Search Graphs

• Any two nodes that root identical subtrees
  (subgraphs) can be merged

33
From Search Trees to Search Graphs

• Any two nodes that root identical subtrees
  (subgraphs) can be merged

34
From Search Trees to Search Graphs

• Any two nodes that root identical
  subtrees/subgraphs can be merged
• Minimal AND/OR search graph closure under merge
  of the AND/OR search tree
• Inconsistent subtrees can be pruned too.
• Some portions can be collapsed or reduced.

35
AND/OR Tree
A
J
A
F
B
B
C
K
C
E
D
E
D
F
G
J
G
H
OR
A
H
K
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
OR
AND
OR
AND
36
An AND/ORgraph
A
J
A
F
B
B
C
K
C
E
D
E
D
F
G
J
G
H
OR
A
H
K
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
0
1
0
1
OR
G
G
J
J
AND
0
1
0
1
0
1
0
1
OR
H
H
H
H
K
K
K
K
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
37
Merging Based on Context

• One way of recognizing nodes that can be merged
• context (X) ancestors of X in pseudo tree that
  are connected to X, or to descendants of X

A
AB
AB
AE
BC
38
AND/OR Search Graph
Constraint Satisfaction Counting Solutions
Context

A
AB
AB
AE
BC
pseudo tree
OR
A
AND
0
1
OR
B
B
AND
1
1
0
0
C
C
OR
C
C
E
E
E
E
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
D
D
D
F
F
F
F
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
context minimal graph
39
AND/OR Tree DFS Algorithm (Belief Updating)
Result P(D1,E0)
Evidence E0
.24408
A
.6
.4
.3028
.1559
0
1
.3028
.1559
B
B
.1
.9
.4
.6
.352
.27
.623
.104
0
1
0
1
.4
.5
.7
.2
.88
.54
.89
.52
E
C
E
C
E
C
E
C
.4
.5
.7
.2
.2
.8
.2
.8
.1
.9
.1
.9
.8
.9
.7
.5
.8
.9
.7
.5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
.8
.9
.7
.7
.8
.9
.5
.5
D
D
D
D
D
D
D
D
.7
.8
.9
.5
.7
.8
.9
.5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR node Marginalization operator (summation)
AND node Combination operator (product)
Value of node updated belief for sub-problem
below
Evidence D1
40
AND/OR Graph DFS Algorithm (Belief Updating)
Result P(D1,E0)
Evidence E0
.24408
A
.6
.4
.3028
.1559
0
1
.3028
.1559
B
B
.1
.9
.4
.6
.352
.27
.623
.104
0
1
0
1
.4
.5
.7
.2
.88
.54
.89
.52
E
C
E
C
E
C
E
C
.4
.5
.7
.2
.2
.8
.2
.8
.1
.9
.1
.9
.8
.9
.7
.5
.8
.9
.7
.5
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
.7
.8
.9
.5
D
D
D
D
.8
.9
.5
.7
0
1
0
1
0
1
0
1
Evidence D1
41
AND/OR context minimal graph
C
0
1
H
H
K
K
0
1
0
1
0
1
0
1
A
A
A
A
L
L
L
L
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
B
B
B
B
B
B
B
B
N
N
N
N
N
N
N
N
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
F
F
F
F
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
0
1
0
1
0
1
0
1
0
1
0
1
P
P
P
P
0
1
0
1
0
1
0
1
P
P
P
P
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
G
G
G
G
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
D
D
D
D
D
D
D
D
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
M
M
M
M
0
1
0
1
0
1
0
1
42
How Big Is the Context?

• Theorem The maximum context size for a pseudo
  tree is equal to the treewidth of the graph along
  the pseudo tree.

43
Treewidth vs. Pathwidth
TREE
ABC
BDFG
G
BDEF
D
A
B
treewidth 3 (max cluster size) - 1
F
EFH
C
E
M
K
H
FHK
L
HJ
KLM
J
CHAIN
ABC
pathwidth 4 (max cluster size) - 1
44
Complexity of AND/OR Graph Search
k domain size n number of variables w
treewidth pw pathwidth
w pw w log n
45
AND/OR search algorithms

• AO(i)
• i the max size of a cache table (i.e. number of
  variables in a context)

i 0
i w treewidth
Space Time
O(n exp w ) O(n exp w )
O( n ) O( exp (w log n) )
46
Searching AND/OR Graphs

• AO(i) searches depth-first, cache i-context
• i the max size of a cache table (i.e. number of
  variables in a context)

i0
iw
i
Space O(exp w) Time O(exp w)
Space O(n) Time O(exp(w log n))
Space O(exp(i) ) Time O(exp(m_ii )
47
Caching
A
J
A
F
B
B
C
K
C
E
D
E
D
F
G
J
G
H
OR
A
H
K
context(D)D context(F)F
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
OR
AND
OR
AND
48
Caching
A
J
A
F
B
B
C
K
C
E
D
E
D
F
context(D)D context(F)F
G
J
G
H
OR
A
H
K
AND
0
1
OR
B
B
AND
0
1
0
1
OR
E
C
E
C
E
C
E
C
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
OR
D
F
D
F
D
F
D
F
D
F
D
F
D
F
D
F
AND
0
1
0
1
OR
G
G
J
J
AND
0
1
0
1
0
1
0
1
OR
H
H
H
H
K
K
K
K
AND
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
49
Size of minimal AND/OR context graphs

• Theorem
• Minimal AND/OR context graph is bounded
  exponentially by its pseudo-tree induced-width.
• The tree-width of a pseudo-chain is path-width
  (pw)
• ? Minimal OR search graph is O(exp(pw)).
• ? Minimal AND/OR graph is O(exp(w))
• Always, w pw, but pwlt w log n

50
OR tree vs. OR graph
OR tree
OR graph
51
AND/OR tree vs. AND/OR graph
AND/OR tree
AND/OR graph
52
All four search spaces
Full OR search tree
Context minimal OR search graph
Full AND/OR search tree
Context minimal AND/OR search graph
53
All four search spaces
Full OR search tree
Context minimal OR search graph
Time-space
Full AND/OR search tree
Context minimal AND/OR search graph
54
Available code

• http//graphmod.ics.uci.edu/group/Software

55
AND/OR w-cutset
56
AND/OR w-cutset
pseudo tree
1-cutset tree
grahpical model
57
Searching AND/OR Graphs

• AO(i) searches depth-first, cache i-context
• i the max size of a cache table (i.e. number of
  variables in a context)

i0
iw
i
Space O(exp w) Time O(exp w)
Space O(n) Time O(exp(w log n))
Space O(exp(i) ) Time O(exp(m_ii )
58
w-cutset Trees Over AND/OR space

• Definition
• T_w is a w-cutset tree relative to backbone tree
  T, iff T_w is roots T and when removed, yields
  tree-width w.
• Theorem
• AO(i) time complexity for backbone T is time
  O(exp(im_i)) and space O(i), m_i is the depth
  of the T_i tree.
• Better than w-cutset O(exp(ic_i)) when c_i is
  the number of nodes in T_i