3. Marginal Independence
• Intuitively: if X ╨ Y, then
– learning that Y=y does not change your belief in X
– and this is true for all values y that Y could take
• For example, weather is marginally independent
of the result of a coin toss
3
5. Conditional Independence
• Intuitively: if X ╨ Y | Z, then
– learning that Y=y does not change your belief in X
when we already know Z=z
– and this is true for all values y that Y could take
and all values z that Z could take
• For example,
5 ExamGrade
╨ AssignmentGrade | UnderstoodMaterial
7. “…probability theory is more fundamentally concerned with
the structure of reasoning and causation than with
numbers.”
Glenn Shafer and Judea Pearl
Introduction to Readings in Uncertain Reasoning,
Morgan Kaufmann, 1990
8. Bayesian Network Motivation
• We want a representation and reasoning system that is based on
conditional (and marginal) independence
– Compact yet expressive representation
– Efficient reasoning procedures
• Bayesian (Belief) Networks are such a representation
– Named after Thomas Bayes (ca. 1702 –1761)
– Term coined in 1985 by Judea Pearl (1936 – )
– Their invention changed the primary focus of AI from logic to probability!
Thomas Bayes Judea Pearl
8
9. Bayesian Networks: Intuition
• A graphical representation for a joint probability
distribution
– Nodes are random variables
• Can be assigned (observed) or unassigned (unobserved)
– Arcs are interactions between nodes
• Encode conditional independence
• An arrow from one variable to another indicates direct influence
• Directed arcs between nodes reflect dependence
– A compact specification of full joint distributions
• Some informal examples:
Smoking At Fire
Understood
Sensor
Material
Assignment Exam
Grade Grade Alarm
9
10. Example of a simple Bayesian
network A B
p(A,B,C) = p(A)p(B) p(C|A,B)
C
• Probability model has simple factored form
• Directed edges => direct dependence
• Absence of an edge => conditional independence
• Also known as belief networks, graphical models, causal networks
• Other formulations, e.g., undirected graphical models
12. Bayesian Networks: Definition
• Discrete Bayesian networks:
– Domain of each variable is finite
– Conditional probability distribution is a conditional probability
table
– We will assume this discrete case
• But everything we say about independence (marginal & conditional)
12
carries over to the continuous case
13. Examples of 3-way Bayesian
Networks
A B C Marginal Independence:
p(A,B,C) = p(A) p(B) p(C)
14. Examples of 3-way Bayesian
Networks
Conditionally independent effects:
p(A,B,C) = p(B|A)p(C|A)p(A)
A B and C are conditionally independent
Given A
e.g., A is a disease, and we model
B C B and C as conditionally independent
symptoms given A
15. Examples of 3-way Bayesian
Networks
A B Independent Causes:
p(A,B,C) = p(C|A,B)p(A)p(B)
C
“Explaining away” effect:
Given C, observing A makes B less likely
e.g., earthquake/burglary/alarm example
A and B are (marginally) independent
but become dependent once C is known
16. Examples of 3-way Bayesian
Networks
A B C Markov dependence:
p(A,B,C) = p(C|B) p(B|A)p(A)
17. Example: Burglar Alarm
• I have a burglar alarm that is sometimes set off by minor
earthquakes. My two neighbors, John and Mary, promised to call
me at work if they hear the alarm
– Example inference task: suppose Mary calls and John doesn’t call. What is
the probability of a burglary?
• What are the random variables?
– Burglary, Earthquake, Alarm, JohnCalls, MaryCalls
18. Example
5 binary variables:
B = a burglary occurs at your house
E = an earthquake occurs at your house
A = the alarm goes off
J = John calls to report the alarm
M = Mary calls to report the alarm
What is P(B | M, J) ? (for example)
We can use the full joint distribution to answer this question
Requires 25 = 32 probabilities
Can we use prior domain knowledge to come up with a Bayesian
network that requires fewer probabilities?
What are the direct influence relationships?
A burglary can set the alarm off
An earthquake can set the alarm off
The alarm can cause Mary to call
The alarm can cause John to call
20. Conditional probability
distributions
• To specify the full joint distribution, we need to specify a
conditional distribution for each node given its parents:
P (X | Parents(X))
Z1 Z2 … Zn
X
P (X | Z1, …, Zn)
22. The joint probability distribution
• For each node Xi, we know P(Xi | Parents(Xi))
• How do we get the full joint distribution P(X1, …, Xn)?
• Using chain rule:
n n
P ( X 1 , , X n ) = ∏ P( X i | X 1 , , X i −1 ) = ∏ P( X i | Parents( X i ) )
i =1 i =1
• For example, P(j, m, a, ¬b, ¬e)
= P(¬b) P(¬e) P(a | ¬b, ¬e) P(j | a) P(m | a)
•
23. Constructing a Bayesian Network:
Step 1
• Order the variables in terms of causality (may be a partial order)
e.g., {E, B} -> {A} -> {J, M}
• P(J, M, A, E, B) = P(J, M | A, E, B) P(A| E, B) P(E, B)
~ P(J, M | A) P(A| E, B) P(E) P(B)
~ P(J | A) P(M | A) P(A| E, B) P(E) P(B)
These CI assumptions are reflected in the graph structure of the
Bayesian network
24. Constructing this Bayesian
Network: Step 2
• P(J, M, A, E, B) =
P(J | A) P(M | A) P(A | E, B) P(E) P(B)
• There are 3 conditional probability tables (CPDs) to be determined:
P(J | A), P(M | A), P(A | E, B)
– Requiring 2 + 2 + 4 = 8 probabilities
• And 2 marginal probabilities P(E), P(B) -> 2 more probabilities
• 2 + 2 + 4 + 1 + 1 = 10 numbers (vs. 25-1 = 31)
• Where do these probabilities come from?
– Expert knowledge
– From data (relative frequency estimates)
– Or a combination of both
25. Number of Probabilities in
Bayesian Networks
• Consider n binary variables
• Unconstrained joint distribution requires O(2n) probabilities
• If we have a Bayesian network, with a maximum of k parents
for any node, then we need O(n 2k) probabilities
• Example
– Full unconstrained joint distribution
• n = 30: need 109 probabilities for full joint distribution
– Bayesian network
• n = 30, k = 4: need 480 probabilities
26. Constructing Bayesian networks
1. Choose an ordering of variables X1, … , Xn
2. For i = 1 to n
– add Xi to the network
– select parents from X1, … ,Xi-1 such that
P(Xi | Parents(Xi)) = P(Xi | X1, ... Xi-1)
29. Example
• Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)?
P(A | J, M) = P(A | J)?
P(A | J, M) = P(A | M)?
30. Example
• Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
31. Example
• Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)?
P(B | A, J, M) = P(B | A)?
32. Example
• Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)? No
P(B | A, J, M) = P(B | A)? Yes
33. Example
• Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)? No
P(B | A, J, M) = P(B | A)? Yes
P(E | B, A ,J, M) = P(E)?
P(E | B, A, J, M) = P(E | A, B)?
34. Example
• Suppose we choose the ordering M, J, A, B, E
P(J | M) = P(J)? No
P(A | J, M) = P(A)? No
P(A | J, M) = P(A | J)? No
P(A | J, M) = P(A | M)? No
P(B | A, J, M) = P(B)? No
P(B | A, J, M) = P(B | A)? Yes
P(E | B, A ,J, M) = P(E)? No
P(E | B, A, J, M) = P(E | A, B)?Yes
35. Example contd.
• Deciding conditional independence is hard in noncausal directions
– The causal direction seems much more natural
• Network is less compact: 1 + 2 + 4 + 2 + 4 = 13 numbers needed
36. A more realistic Bayes Network:
Car diagnosis
• Initial observation: car won’t start
• Orange: “broken, so fix it” nodes
• Green: testable evidence
• Gray: “hidden variables” to ensure sparse structure, reduce parameteres
38. Given a graph, can we “read off” conditional independencies?
39. Are there wrong network
structures?
• Some variable orderings yield more compact, some less
compact structures
– Compact ones are better
– But all representations resulting from this process are correct
– One extreme: the fully connected network is always correct
but rarely the best choice
• How can a network structure be wrong?
– If it misses directed edges that are required
40. Summary
• Bayesian networks provide a natural representation for
(causally induced) conditional independence
• Topology + conditional probability tables
• Generally easy for domain experts to construct
41. Probabilistic inference
• A general scenario:
– Query variables: X
– Evidence (observed) variables: E = e
– Unobserved variables: Y
• If we know the full joint distribution P(X , E, Y), how can we
perform inference about X?
P( X , e)
P( X | E = e) = ∝ ∑ y P( X , e, y )
P (e )
• Problems
– Full joint distributions are too large
– Marginalizing out Y may involve too many summation terms
42. Conclusions…
• Full joint distributions are intractable to work with
– Conditional independence assumptions allow us to model real-
world phenomena with much simpler models
– Bayesian networks are a systematic way to construct
parsimonious structured distributions
• How do we do inference (reasoning) in Bayesian
networks?
– Systematic algorithms exist
– Complexity depends on the structure of the graph
![Today’s Lecture
• Recap: Joint distribution, independence, marginal
independence, conditional independence
• Bayesian networks
• Reading:
– Sections 14.1-14.4 in AIMA [Russel & Norvig]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)