1. UIUC CS 497: Section EA Lecture #6
Reasoning in Artificial Intelligence
Professor: Eyal Amir
Spring Semester 2004
(Based on slides by Lise Getoor and Alvaro Cardenas (UMD)
(in turn based on slides by Nir Friedman (Hebrew U)))

2. Last Time Tree decomposition in first-order logic Applications:
Provably better computational bounds when low treewidth in propositional logic
Eliminate possible interactions between clauses
Applications:
Planning, spatial reasoning

3. Today Probabilistic graphical models Treewidth methods:
Variable elimination
Clique tree algorithm
Applications du jour: Sensor Networks

4. Independent Random Variables
Two variables X and Y are independent if
P(X = x|Y = y) = P(X = x) for all values x,y
That is, learning the values of Y does not change prediction of X
If X and Y are independent then
P(X,Y) = P(X|Y)P(Y) = P(X)P(Y)
In general, if X1,…,Xp are independent, then P(X1,…,Xp)= P(X1)...P(Xp)
Requires O(n) parameters

5. Conditional Independence
Unfortunately, most of random variables of interest are not independent of each other
A more suitable notion is that of conditional independence
Two variables X and Y are conditionally independent given Z if
P(X = x|Y = y,Z=z) = P(X = x|Z=z) for all values x,y,z
That is, learning the values of Y does not change prediction of X once we know the value of Z
notation: I ( X , Y | Z )

6. Example: Family trees Noisy stochastic process: Example: Pedigree
Homer
Bart
Marge
Lisa
Maggie
Noisy stochastic process:
Example: Pedigree
A node represents an individual’s genotype
Modeling assumptions:
Ancestors can effect descendants' genotype only by passing genetic materials through intermediate generations

7. Markov Assumption
Ancestor X Y1 Y2
Non-descendent
We now make this independence assumption more precise for directed acyclic graphs (DAGs)
Each random variable X, is independent of its non-descendents, given its parents Pa(X)
Formally, I (X, NonDesc(X) | Pa(X))
Parent
Non-descendent
Descendent

8. Markov Assumption Example
Earthquake
Radio
Burglary
Alarm
Call
In this example:
I ( E, B )
I ( B, {E, R} )
I ( R, {A, B, C} | E )
I ( A, R | B,E )
I ( C, {B, E, R} | A)

9. I-Maps
A DAG G is an I-Map of a distribution P if all Markov assumptions implied by G are satisfied by P
(Assuming G and P both use the same set of random variables)
Examples: X Y X Y

10. Factorization
Given that G is an I-Map of P, can we simplify the representation of P?
Example:
Since I(X,Y), we have that P(X|Y) = P(X)
Applying the chain rule P(X,Y) = P(X|Y) P(Y) = P(X) P(Y)
Thus, we have a simpler representation of P(X,Y) X Y

11. Factorization Theorem
Thm: if G is an I-Map of P, then
Proof:
By chain rule:
wlog. X1,…,Xp is an ordering consistent with G
From assumption:
Since G is an I-Map, I (Xi, NonDesc(Xi)| Pa(Xi))
Hence,
We conclude, P(Xi | X1,…,Xi-1) = P(Xi | Pa(Xi) )

12. Factorization Example
Earthquake
Radio
Burglary
Alarm
Call
P(C,A,R,E,B) = P(B)P(E|B)P(R|E,B)P(A|R,B,E)P(C|A,R,B,E)
versus
P(C,A,R,E,B) = P(B) P(E) P(R|E) P(A|B,E) P(C|A)

13. Consequences  each conditional probability can be specified compactly
We can write P in terms of “local” conditional probabilities
If G is sparse,
that is, |Pa(Xi)| < k ,
 each conditional probability can be specified compactly
e.g. for binary variables, these require O(2k) params.
 representation of P is compact
linear in number of variables

14. Summary We defined the following concepts
The Markov Independences of a DAG G
I (Xi , NonDesc(Xi) | Pai )
G is an I-Map of a distribution P
If P satisfies the Markov independencies implied by G
We proved the factorization theorem
if G is an I-Map of P, then

15. Conditional Independencies
Let Markov(G) be the set of Markov Independencies implied by G
The factorization theorem shows
G is an I-Map of P 
We can also show the opposite:
Thm:
 G is an I-Map of P

16. Proof (Outline) X Z
Example: Y

17. Implied Independencies
Does a graph G imply additional independencies as a consequence of Markov(G)?
We can define a logic of independence statements
Some axioms:
I( X ; Y | Z )  I( Y; X | Z )
I( X ; Y1, Y2 | Z )  I( X; Y1 | Z )

18. d-seperation
A procedure d-sep(X; Y | Z, G) that given a DAG G, and sets X, Y, and Z returns either yes or no
Goal:
d-sep(X; Y | Z, G) = yes iff I(X;Y|Z) follows from Markov(G)

19. Paths Intuition: dependency must “flow” along paths in the graph
A path is a sequence of neighboring variables
Examples:
R  E  A  B
C  A  E  R
Earthquake
Radio
Burglary
Alarm
Call

20. Paths We want to know when a path is
active -- creates dependency between end nodes
blocked -- cannot create dependency end nodes
We want to classify situations in which paths are active.

21. Path Blockage Blocked Unblocked Blocked Active Three cases: E R A E R
Common cause
Blocked
Unblocked
Blocked
Active E R A E R A

22. Path Blockage Blocked Active Blocked Unblocked Three cases: E A C
Common cause
Intermediate cause
Blocked
Active
Blocked
Unblocked E C A

23. Path Blockage Blocked Active Blocked Unblocked Three cases: E B A C E
Common cause
Intermediate cause
Common Effect
Blocked
Active
Blocked
Unblocked E B A C E B A C

24. Path Blockage -- General Case
A path is active, given evidence Z, if
Whenever we have the configuration B or one of its descendents are in Z
No other nodes in the path are in Z
A path is blocked, given evidence Z, if it is not active. A C B

25. Example
d-sep(R,B)? E B R A C

26. Example
d-sep(R,B) = yes
d-sep(R,B|A)? E B R A C

27. Example
d-sep(R,B) = yes
d-sep(R,B|A) = no
d-sep(R,B|E,A)? E B R A C

28. d-Separation
X is d-separated from Y, given Z, if all paths from a node in X to a node in Y are blocked, given Z.
Checking d-separation can be done efficiently (linear time in number of edges)
Bottom-up phase: Mark all nodes whose descendents are in Z
X to Y phase: Traverse (BFS) all edges on paths from X to Y and check if they are blocked

29. Soundness
Thm: If
G is an I-Map of P
d-sep( X; Y | Z, G ) = yes
then
P satisfies I( X; Y | Z )
Informally: Any independence reported by d-separation is satisfied by underlying distribution

30. Completeness Thm: If d-sep( X; Y | Z, G ) = no
then there is a distribution P such that
G is an I-Map of P
P does not satisfy I( X; Y | Z )
Informally: Any independence not reported by d-separation might be violated by the underlying distribution
We cannot determine this by examining the graph structure alone

31. Summary: Structure
We explored DAGs as a representation of conditional independencies:
Markov independencies of a DAG
Tight correspondence between Markov(G) and the factorization defined by G
d-separation, a sound & complete procedure for computing the consequences of the independencies
Notion of minimal I-Map
P-Maps
This theory is the basis for defining Bayesian networks

32. Inference
We now have compact representations of probability distributions:
Bayesian Networks
Markov Networks
Network describes a unique probability distribution P
How do we answer queries about P?
We use inference as a name for the process of computing answers to such queries

33. Today Probabilistic graphical models Treewidth methods:
Variable elimination
Clique tree algorithm
Applications du jour: Sensor Networks

34. Queries: Likelihood There are many types of queries we might ask.
Most of these involve evidence
An evidence e is an assignment of values to a set E variables in the domain
Without loss of generality E = { Xk+1, …, Xn }
Simplest query: compute probability of evidence
This is often referred to as computing the likelihood of the evidence

35. Queries: A posteriori belief
Often we are interested in the conditional probability of a variable given the evidence
This is the a posteriori belief in X, given evidence e
A related task is computing the term P(X, e)
i.e., the likelihood of e and X = x for values of X

36. A posteriori belief This query is useful in many cases:
Prediction: what is the probability of an outcome given the starting condition
Target is a descendent of the evidence
Diagnosis: what is the probability of disease/fault given symptoms
Target is an ancestor of the evidence
the direction between variables does not restrict the directions of the queries

37. Queries: MAP
In this query we want to find the maximum a posteriori assignment for some variable of interest (say X1,…,Xl )
That is, x1,…,xl maximize the probability P(x1,…,xl | e)
Note that this is equivalent to maximizing P(x1,…,xl, e)

38. Queries: MAP We can use MAP for: Classification Explanation
find most likely label, given the evidence
Explanation
What is the most likely scenario, given the evidence

39. Complexity of Inference
Thm:
Computing P(X = x) in a Bayesian network is NP-hard
Not surprising, since we can simulate Boolean gates.

40. Approaches to inference
Exact inference
Inference in Simple Chains
Variable elimination
Clustering / join tree algorithms
Approximate inference – next time
Stochastic simulation / sampling methods
Markov chain Monte Carlo methods
Mean field theory – your presentation

41. Variable Elimination General idea: Write query in the form Iteratively
Move all irrelevant terms outside of innermost sum
Perform innermost sum, getting a new term
Insert the new term into the product

42. Example “Asia” network: Visit to Asia Smoking Lung Cancer Tuberculosis
Abnormality
in Chest
Bronchitis
X-Ray
Dyspnea

43. Need to eliminate: v,s,x,t,l,a,b Initial factors
We want to compute P(d)
Need to eliminate: v,s,x,t,l,a,b
Initial factors
“Brute force approach”
Complexity is exponential in the size of the graph (number of variables) = T. N=number of states for each variable

44. Need to eliminate: v,s,x,t,l,a,b Initial factors
We want to compute P(d)
Need to eliminate: v,s,x,t,l,a,b
Initial factors
Eliminate: v
Compute:
Note: fv(t) = P(t)
In general, result of elimination is not necessarily a probability term

45. Need to eliminate: s,x,t,l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: s,x,t,l,a,b
Initial factors
Eliminate: s
Compute:
Summing on s results in a factor with two arguments fs(b,l)
In general, result of elimination may be a function of several variables

46. Need to eliminate: x,t,l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: x,t,l,a,b
Initial factors
Eliminate: x
Compute:
Note: fx(a) = 1 for all values of a !!

47. Need to eliminate: t,l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: t,l,a,b
Initial factors
Eliminate: t
Compute:

48. We want to compute P(d) Need to eliminate: l,a,b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: l,a,b
Initial factors
Eliminate: l
Compute:

49. We want to compute P(d) Need to eliminate: b Initial factorsV S L T A B X D
We want to compute P(d)
Need to eliminate: b
Initial factors
Eliminate: a,b
Compute:

50. Different elimination ordering: Need to eliminate: a,b,x,t,v,s,l
Initial factors
Intermediate factors:
Complexity is exponential in the size of the factors!

51. Variable Elimination
We now understand variable elimination as a sequence of rewriting operations
Actual computation is done in elimination step
Exactly the same computation procedure applies to Markov networks
Computation depends on order of elimination

52. Markov Network (Undirected Graphical Models)
A graph with hyper-edges (multi-vertex edges)
Every hyper-edge e=(x1…xk) has a potential function fe(x1…xk)
The probability distribution is

53. Complexity of variable elimination
Suppose in one elimination step we compute
This requires
multiplications
For each value for x, y1, …, yk, we do m multiplications
additions
For each value of y1, …, yk , we do |Val(X)| additions
Complexity is exponential in number of variables in the intermediate factor

54. Undirected graph representation
At each stage of the procedure, we have an algebraic term that we need to evaluate
In general this term is of the form: where Zi are sets of variables
We now plot a graph where there is undirected edge X--Y if X,Y are arguments of some factor
that is, if X,Y are in some Zi
Note: this is the Markov network that describes the probability on the variables we did not eliminate yet

55. Chordal Graphs elimination ordering  undirected chordal graph Graph:
Maximal cliques are factors in elimination
Factors in elimination are cliques in the graph
Complexity is exponential in size of the largest clique in graph V S L T A B X D L T A B X V S D

56. Induced Width
The size of the largest clique in the induced graph is thus an indicator for the complexity of variable elimination
This quantity is called the induced width of a graph according to the specified ordering
Finding a good ordering for a graph is equivalent to finding the minimal induced width of the graph

57. PolyTrees
A polytree is a network where there is at most one path from one variable to another
Thm:
Inference in a polytree is linear in the representation size of the network
This assumes tabular CPT representation A C B D E F G H

58. Today Probabilistic graphical models Treewidth methods:
Variable elimination
Clique tree algorithm
Applications du jour: Sensor Networks

59. Junction Tree Why junction tree? Objective
More efficient for some tasks than variable elimination
We can avoid cycles if we turn highly-interconnected subsets of the nodes into “supernodes”  cluster
Objective
Compute
is a value of a variable and is evidence for a set of variable

60. Properties of Junction Tree
An undirected tree
Each node is a cluster (nonempty set) of variables
Running intersection property:
Given two clusters and , all clusters on the path between and contain
Separator sets (sepsets):
Intersection of the adjacent cluster
ADE
ABD
DEF AD DE
Cluster ABD
Sepset DE

61. Potentials Potentials: Marginalization Multiplication Denoted by
, the marginalization of into X
Multiplication
, the multiplication of and

62. Properties of Junction Tree
Belief potentials:
Map each instantiation of clusters or sepsets into a real number
Constraints:
Consistency: for each cluster and neighboring sepset
The joint distribution

63. Properties of Junction Tree
If a junction tree satisfies the properties, it follows that:
For each cluster (or sepset) ,
The probability distribution of any variable , using any cluster (or sepset) that contains

64. Building Junction Trees
DAG
Moral Graph
Triangulated Graph
Identifying Cliques
Junction Tree

65. Constructing the Moral GraphB D C E G F H

66. Constructing The Moral Graph
Add undirected edges to all co-parents which are not currently joined –Marrying parents A B D C E G F H

67. Constructing The Moral Graph
Add undirected edges to all co-parents which are not currently joined –Marrying parents
Drop the directions of the arcs A B D C E G F H

68. Triangulating
An undirected graph is triangulated iff every cycle of length >3 contains an edge to connects two nonadjacent nodes A B D C E G F H

69. Identifying Cliques
A clique is a subgraph of an undirected graph that is complete and maximal A B D C E G F H
EGH
ADE
ABD
ACE
DEF
CEG

70. Junction Tree A junction tree is a subgraph of the clique graph that
is a tree
contains all the cliques
satisfies the running intersection property
EGH
ADE
ABD
ACE
DEF
CEG
ADE
ABD
ACE AD AE
CEG CE
DEF DE
EGH EG

71. Principle of Inference
DAG
Junction Tree
Inconsistent Junction Tree
Initialization
Consistent Junction Tree
Propagation
Marginalization

72. Example: Create Join Tree
HMM with 2 time steps: X1 X2 Y1 Y2
Junction Tree:
X1,X2
X1,Y1
X2,Y2 X1 X2

73. Example: Initialization
X1,X2
X1,Y1
X2,Y2 X1 X2
Variable
Associated Cluster
Potential function X1
X1,Y1 Y1 X2
X1,X2 Y2
X2,Y2

74. Example: Collect Evidence
Choose arbitrary clique, e.g. X1,X2, where all potential functions will be collected.
Call recursively neighboring cliques for messages:
1. Call X1,Y1.
1. Projection:
2. Absorption:

75. Example: Collect Evidence (cont.)
2. Call X2,Y2:
1. Projection:
2. Absorption:
X1,X2
X1,Y1
X2,Y2 X1 X2

76. Example: Distribute Evidence
Pass messages recursively to neighboring nodes
Pass message from X1,X2 to X1,Y1:
1. Projection:
2. Absorption:

77. Example: Distribute Evidence (cont.)
Pass message from X1,X2 to X2,Y2:
1. Projection:
2. Absorption:
X1,X2
X1,Y1
X2,Y2 X1 X2

78. Example: Inference with evidence
Assume we want to compute: P(X2|Y1=0,Y2=1) (state estimation)
Assign likelihoods to the potential functions during initialization:

79. Example: Inference with evidence (cont.)
Repeating the same steps as in the previous case, we obtain:

80. Next Time Approximate Probabilistic Inference via sampling Gibbs
Priority
MCMC

81. THE END

82. Example: Naïve Bayesian Model
A common model in early diagnosis:
Symptoms are conditionally independent given the disease (or fault)
Thus, if
X1,…,Xp denote whether the symptoms exhibited by the patient (headache, high-fever, etc.) and
H denotes the hypothesis about the patients health
then, P(X1,…,Xp,H) = P(H)P(X1|H)…P(Xp|H),
This naïve Bayesian model allows compact representation
It does embody strong independence assumptions

83. Elimination on Trees
Formally, for any tree, there is an elimination ordering with induced width = 1
Thm
Inference on trees is linear in number of variables