On Mixed Probabilistic and deterministic Networks

1
On Mixed Probabilistic and deterministic Networks

• COMPSCI 276

2
Overview

• Introduction for Graphical models (Constraint
  networks, probabilistic networks and Influence
  diagrams)
• Inference and search methods for GM
• Mixed networks.
• Inference algorithms for mixed networks
• Search algorithms for mixed networks
• Solving constraints under Uncertainty.

3
When Do we Need Both Constraints and
probabilities

• Constraint problems (e.g. scheduling, design,
  resource allocation) with partial knowledge of
  the hard constraints.
• Need to express uncertainty of parameters
• Change of constraints over time
• Probabilistic networks modeling agents behavior
  in the world.
• Requires adding constraints to world knowledge
• Express complex queries (cnfs)
• Express complex evidence

4
Example Modeling Students Class Scheduling.

• Constraints
• Course prerequisites
• Requirements for students major
• Class size limits
• A student can be only at a single place at a time
• Probabilistic information
• Liklihood of students grade in the class, given
  past performance
• The liklihood a class will be given next semester
  if it is given now
• The liklihood a student will take a class that is
  taught by a popular teacher
• Queries
• Belief updating P(class-size given that the
  teacher is John)
• Mpe What is the most likely scenario of class
  taking next year? (mpe)
• MAP What is the liklihhod that all students can
  take all required classes?
• Given some cost of operating classes, What is the
  minimum expected cost?

5
Frameworks for Reasoning

• Deterministic Constraint Networks
• Center around humans actions/decisions
  (scheduling, planning, design)
• Probabilistic Bayesian Networks
• Centers on describing the word (circuits, linkage
  analysis, diagnosis)
• Influence Diagrams
• Distinguishes decision and chance variables.
• Modeling agents behavior in the environment
  requires both Probabilities and constraints.

6
Propositional Reasoning
Example party problem

• If Alex goes, then Becky goes
• If Chris goes, then Alex goes
• Question
• Is it possible that Chris goes to the party
  but Becky does not?

7
Probabilistic Reasoning
Party example the weather effect

• Alex is-likely-to-go in bad weather
• Chris rarely-goes in bad weather
• Becky is indifferent but unpredictable
• Questions
• Given bad weather, which group of individuals is
  most likely to show up at the party?
• What is the probability that Chris goes to the
  party but Becky does not?

P(W,A,C,B) P(BW) P(CW) P(AW)
P(W) P(A,C,BWbad) 0.9 0.1 0.5
8
Constraint Networks
A
B
C

• Constraint Satisfaction
• Find one/all solutions/Counting
• Max-CSP
• Find a solution that violates the least number of
  constraints
• Constraint optimization
• Given cost function find optimal solution

D
F
G
9
Bucket EliminationVariable elimination

Bucket E E ¹ D, E ¹ C Bucket D D ¹
A Bucket C C ¹ B Bucket B B ¹ A Bucket A
10
The Idea of Conditioning
11
Overview

• Introduction for Graphical models (Constraint
  networks, probabilistic networks and Influence
  diagrams)
• Inference and search methods for GM
• Mixed networks
• Inference algorithms for mixed networks
• Search algorithms for mixed networks
• Cuurent state of the art.

12
Mixed Networks

• Augmenting Probabilistic networks with
  constraints because
• Some information in the world is deterministic
  and undirected (X not-eq Y)
• Some queries are complex or evidence are complex
  (cnfs)
• Queries are probabilistic queries

13
Two Loci Inheritance
Recombinant
14
Bayesian Network for Recombination
L11m
L11f
L12f
L12m
Locus 1
S13m
X11
S13f
X12
y2
y1
L13f
L13m
X13
y3
L21f
L21m
L22f
L22m
S23m
X21
S23f
X22
Locus 2
L23m
L23f
P(eT) ?
X23
15
6 people, 3 markers
L12m
L12f
L11m
L11f
X12
X11
S15m
S13m
L13m
L13f
L14m
L14f
X14
X13
S15m
S15m
L15m
L15f
L16m
L16f
S16m
S15m
X15
X16
L22m
L22f
L21m
L21f
X21
X22
S25m
S23m
L23m
L23f
L24m
L24f
X23
X24
S25m
S25m
L25m
L25f
L26m
L26f
S26m
S25m
X25
X26
L32m
L32f
L31m
L31f
X32
X31
S35m
S33m
L33m
L33f
L34m
L34f
X33
X34
S35m
S35m
L35m
L35f
L36m
L36f
S36m
S35m
X35
X36
16
Constraints as CPTs

• Express each constraint as a probability table
  using a new child variable.
• XY Z expressed as
• P(H X,Y,Z) 1 iff Z XY

H1
H
P( ? ZXY)
X
Y
Z
X
Y
Z
17
Hiding constraints as probabilities is
undesirable because

• Not natural, adds many variables

18
Hiding constraints as probabilities is
undesirable because

• Not natural, adds many variables

19
Hiding constraints as probabilities is
undesirable because

• Not natural, adds many variables
• Semantic is not clear confuses causal and
  non-causal informatio
• Most important looses computational power of
  constraints constraint propagation, local
  inference
• But, it is feasible in principle!!!

20
Party example again
PN
CN
Query Is it likely that Chris goes to the party
if Becky does not but the weather is bad?
Semantics? Algorithms?
21
Mixed Networks
Constraint Networks

• Belief Networks

A
A
F
F
B
C
B
C
D
E
D
E
B
R
22
The Auxiliary Network
A
A
F
F
B
C
B
C
D
E
D
E
A
X1
F
B
C
X4
X2
D
E
X3

• Drawbacks
• constraint information is not readily
  exploitable
• clutters the problem structure

23
Mixed Networks
A
A
F
F
B
C
B
C
Moral mixed graph
D
E
D
E
M(B,R)
24
Query Processing
25
Equivalent Mixed Networks

• The probability distribution PM is uniquely
  defined by PB and ?
• Constraint propagation techniques can be used to
  eliminate inconsistent tuples, yielding an easier
  computation on the belief part of the mixed
  network

26
Mixed Graph

• The mixed graph is the union of the belief
  network graph and the constraint network graph
• The moral mixed graph is the union of the moral
  graph of the belief network and the graph of the
  constraint network

27
Tasks for Mixed Networks

• Given M (B,R)
• Belief updating Find the liklihood of X given e
  and assuming consistency (P(xR,e) ?)
• MPE find probability of most likely solution of
  R
• MAP Find the probability of most likely
  consistent assignment to a subset of variables
• Constraint Probability Evaluation (CPE) Find the
  probability that an assignment is consistent,
• All these can be extended to influence diagrams,
  and
• We can add constraints between control variables
• The relevant probability distribution is
  conditional of R

28
Overview

• Introduction
• Graphical models Constraint networks,
  probabilistic networks and Influence diagrams
• Mixed networks, queries.
• Inference for mixed networks
• Processing Mixed networks by Search
• Curent state of the art.

29
Bucket Elimination for Mixed Networks Computing
Probability of a cnf query
Bucket G P(GF,D) Bucket D
P(DA,B) Bucket B P(BA) P(FB,C) Bucket
C P(CA) Bucket F Bucket A
A
C
B
F
D
G
Belief network P(g,f,d,c,b,a) P(gf,d)P(fc,b)P(d
b,a)P(ba)P(ca)P(a)
30
Complexity

• Time and space exponential in the induced-width
  of the mixed graph, whose evidence node and unit
  literals are removed.
• Apply constraint propagation to the constraint
  portion and then solve the mixed network.

31
Elim-CPE-D on Mildew Network
20 instances with Mildew network. 5 constraints,
relation arity 2, relation tightness 34 percent,
5 evidence nodes.
32
Elim-CPE-D on Insurance network(Dechter and
Larkin, UAI2001)
19 instances with Insurance network.
20 relations, arity 3, tightness 25 , 5
evidence nodes.
33
Overview

• Introduction
• Graphical models Constraint networks,
  probabilistic networks and Influence diagrams
• Mixed networks, queries.
• Inference for mixed networks
• Search for Mixed networks
• Curent state of the art.

34
AND/OR search tree

• The AND/OR search tree of a constraint network R
  relative to a pseudo-tree, T, has alternating
  levels of AND nodes (variables) and OR nodes
  (values)
• The root is the root of T (OR node)
• Successor function
• The successors of an OR node X are all its
  consistent values along its path
• The successors of an AND node ltX,vgt are all the
  children of X in T
• AND nodes have labels
• A solution is a subtree

35
AND/OR search tree properties

• 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 A constraint network that has a
  tree-width w has an AND/OR search tree whose
  size is bounded by O(exp(w log n)) (similar to
  RC, Darwiche 01 Bacchus et.al 03 Freuder 85
  Bayardo 95)
• Result AND/OR search tree algorithms are
• Space O(n)
• Time O(exp(w log n))

k domain size m pseudo-tree depth n
number of variables
36
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 subproblem
below
Evidence D1
37
The Effect of Constraint Propagation
Domains are 1,2,3,4
CONSTRAINTS ONLY
FORWARD CHECKING
MAINTAINING ARC CONSISTENCY
38
Experimental results

• Parameters of the mixed networks
• N number of variables
• K number of values per variable

• Belief network - (N,K,R,P)
• R number of root nodes
• P number of parents
• Measures
• Time
• Number of nodes
• Number of dead-ends

• Constraint network - (N,K,C,S,t)
• C number of constraints
• S scope size of the constraints
• t tightness (no. of allowed tuples)
• Algorithms
• AO-C constraint checking only
• AO-FC forward checking
• AO-RFC relational forward checking

OR vs. AND/OR Spaces
39
AND/OR Search Algorithms (1)
40
AND/OR Search Algorithms (2)
41
AND/OR Search vs. Bucket Elimination
42
Constrained Optimization
Example power plant scheduling
43
Adding Uncertainty to Constraint Optimization

• Uncertainty in the constraint parameters
• X1x2 gt d, d random variable
• Model d as a chance variable, x1 and x2 as
  control variables in influence diagrams
• And seek an assignment that maximizes the
  expected cost.

44
Conclusion

• Mixed networks combine belief and constraint
  networks. The new formalism borrows specific
  strengths from both
• AND/OR search spaces are always more effective
  than traditional OR spaces
• Linear space search as well as caching algorithms
  can be used with mixed networks
• Benefits of mixed networks
• Constraint propagation techniques can be applied
  straightforwardly, maintaining their properties
  of convergence and fixed point
• The semantics is much clearer by separating
  probabilistic and deterministic information
• The algorithms can be made more efficient

45
Mixed Beliefs and Constraints

• Assume the CN is a cnf formula
• Queries over hybrid network
• Complex evidence structure
• All reduce to CNF queries over a Belief network
• CPE (CNF probability evaluation) Given a belief
  network, and a cnf, find its probability.

46
How to process mixed network?
B
A
F
C
S
D
E
47
A Hybrid Belief Network
Bucket G P(GF,D) Bucket F P(FB,C)
Bucket D P(DA,B) Bucket C
P(CA) Bucket B P(BA) Bucket A P(A)
Belief network P(g,f,d,c,b,a) P(gf,d)P(fc,b)P(d
b,a)P(ba)P(ca)P(a)
48
Variable elimination for a mixed network
Bucket G P(GF,D) Bucket F P(FB,C)
Bucket D P(DA,B) Bucket C
P(CA) Bucket B P(BA) Bucket A P(A)
Bucket G P(GF,D) Bucket F P(FB,C)
Bucket D P(DA,B) Bucket C
P(CA) Bucket B P(BA) Bucket A P(A)
(b) Elim-CPE-D with clause extraction
(a) regular Elim-CPE
49
Trace of Elim-CPE
Bucket G P(GF,D) Bucket D
P(DA,B) Bucket B P(BA) P(FB,C) Bucket
C P(CA) Bucket F Bucket A
A
C
B
F
D
G
Belief network P(g,f,d,c,b,a) P(gf,d)P(fc,b)P(d
b,a)P(ba)P(ca)P(a)
50
Linear Space Search - Complexity

• THEOREM Given a graphical model R and a legal
  tree T, its AND/OR search tree ST(R) is sound and
  complete (contains all and only solutions) and
  its size is O(nexp(m)) where m is the legal
  trees depth. A graphical model that has a
  tree-width w has an AND/OR search tree whose
  size is O(exp(w log n)).
• THEOREM The minimal AND/OR search graph is
  bounded exponentially by the primal graphs
  tree-width, O(exp (w)), while the OR minimal
  search graph is bounded exponentially by its
  path-width.
• w pw m w log n
• w tree-width
• pw path-width
• m minimal depth over all legal trees
• n number of variables

51
Constraint checking only
A
OR
AND
1
2
3
4
OR
C
B
C
B
B
B
2
3
4
2
3
4
3
4
3
4
4
AND
E
D
D
E
D
F
F
F
D
D
F
F
D
OR
AND
Domains are 1,2,3,4
3
4
3
4
4
4
4
3
4
4
OR
H
G
G
I
I
G
K
K
K
G
K
AND
4
4
4
4
52
FC and MAC
A
OR
AND
1
2
3
4
OR
C
B
C
B
B
B
Forward Checking
2
3
4
2
3
4
3
4
3
4
4
AND
E
D
D
E
D
F
F
F
D
D
F
F
D
OR
AND
3
4
3
4
4
4
4
3
4
4
OR
H
G
G
I
I
G
K
K
K
G
K
AND
4
4
4
4
A
OR
AND
1
2
3
4
OR
C
B
C
B
B
B
Maintaining Arc Consistency
2
3
4
2
3
4
3
4
3
4
4
AND
E
D
D
E
D
F
F
F
D
D
F
F
D
OR
AND
3
4
3
4
4
4
4
3
4
4
OR
H
G
G
I
I
G
K
K
K
G
K
AND
4
4
4
4