1. Probabilistic Reasoning
Comp3710 Artificial Intelligence
Computing Science
Thompson Rivers University

2. Course Outline Part I – Introduction to Artificial Intelligence
Part II – Classical Artificial Intelligence
Part III – Machine Learning
Introduction to Machine Learning
Neural Networks
Probabilistic Reasoning and Bayesian Belief Networks
Artificial Life: Learning through Emergent Behavior
Part IV – Advanced Topics
Bayesian Networks

3. Learning Objectives
Bayesian Networks

4. Unit Outline
Introduction
Bayesian networks
Summary
Bayesian Networks

5. Topics
1. Introduction
A systematic way, called Bayesian networks, to represent the world probabilistically – independence and conditional independence relationships
Application areas:
consumer help desks,
nuclear reactor diagnosis,
tissue pathology,
pattern recognition,
credit assessment,
computer network diagnosis,
...
Representing knowledge in an uncertain domain
Semantics of Bayesian networks
Example:
Bayesian Networks

6. 2. Bayesian networks Definitions
Example – Weather, Cavity, Catch, and Toothache
Example – Burglary net
How to answer queries in the Burglary net
How to construct Bayesian networks
Bayesian Networks

7. 2.
Definitions
Bayesian network, belief network, probabilistic network, causal network, or also called knowledge map: A simple, graphical notation for conditional independence assertions and hence for compact specification of full joint distributions
Syntax:
a set of nodes, one per random variable
a directed, acyclic graph (link means “a parent node influences its child nodes.”)
a conditional probability distribution for each node given its parents:
P (Xi | Parents (Xi))
In the simplest case, conditional probability distribution represented as a conditional probability table (CPT) giving the distribution over Xi for each combination of parent values
Bayesian Networks

8. Example – Weather, Toothache, C, C2.
Example – Weather, Toothache, C, C
Considering Weather, Toothache, Cavity, and Catch,
Weather is independent of the other variables.
[Q] What else does not get any influence from others?
Cavity directly influences Toothache and Catch.
Toothache and Catch are conditionally independent given Cavity.
Topology of network encodes conditional independence assertions:
Bayesian Networks

9. Example - Burglary
[Q] I'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar?
[Q] Random variables?
Burglary, Earthquake, Alarm, JohnCalls, MaryCalls
Network topology reflects direct "causal" or “influence” knowledge.
[Q] In what way? Can you draw the network then?
A burglar 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.
Bayesian Networks

10. No need to include P(e)
[Q] What is P(Alarm=T)?
P(a) = P(abe) + P(abe) +
P(abe) + P(abe)
= P(a|b,e) P(be) +
P(a|b,e) P(be) +
P(a|b,e) P(be) +
P(a|b,e) P(be)
= ???
P(a | b,e)
P(a | b,e) = ???
John calls to say my alarm is ringing.
[Q] What does it mean?
[Q] What is P(JohnCalls=T)?
Bayesian Networks

11. [Q] What is P(JohnCalls=T)? P(j) = P(j  a) + ??? = ??? + ??? = ???
[Q] What is P(Alarm=T)?
P(a) = P(abe) + P(abe) +
P(abe) + P(abe)
= P(a|b,e) P(be) +
P(a|b,e) P(be) +
P(a|b,e) P(be) +
P(a|b,e) P(be)
= P(a|b,e) P(b) P(e) + …
= ???
P(a | b,e)
P(a | b,e) = ???
John calls to say my alarm is ringing.
[Q] What is P(JohnCalls=T)?
P(j) = P(j  a) + ???
= ??? + ???
= ???
Bayesian Networks

12. [Q] What is P(JohnCalls=T)? P(j) = P(j  a) + P(j  a)
[Q] What is P(Alarm=T)?
P(a) = P(abe) + P(abe) +
P(abe) + P(abe)
= P(a|b,e) P(be) +
P(a|b,e) P(be) +
P(a|b,e) P(be) +
P(a|b,e) P(be)
= P(a|b,e) P(b) P(e) + …
= ???
P(a | b,e)
John calls to say my alarm is ringing.
[Q] What is P(JohnCalls=T)?
P(j) = P(j  a) + P(j  a)
= P(j | a) P(a) + P(j | a) P(a)
= ???
Bayesian Networks

13. 2.
Compactness
A CPT for Boolean Xi with k Boolean parents has 2k rows for the combinations of parent values.
Each row requires one number p for Xi = true (the number for Xi = false is just 1 - p)
If each variable has no more than k parents, the complete network requires O(n · 2k) numbers
I.e., grows linearly with n, vs. O(2n) for the full joint distribution
But, for burglary net, = 10 numbers (vs = 31)
Bayesian Networks

14. How to answer queries in Burglary net
P(X1, …, Xn) = Πi = 1 P(Xi | Parents(Xi))
[Q] P(j  m  a  b  e) = ???
= .9 * .7 * .001 * .999 * .998 ≈ n
Bayesian Networks

15. P(X1, …, Xn) =  P(Xi | Parents(Xi))
[Q] I'm at work, neighbor John calls to say my alarm is ringing, but neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. What is the chance that the alarm was ringing and there is a burglar?
P(j  m  a  b) = ??? [Q] How to deal with Earthquake?
= P(j | a) P(m | a) P(a | b,e) P(b) P(e) + P(j | a) P(m | a) (P(a | b,e) P(b) P(e)
= P(j | a) (1 - P(m | a)) P(a | b,e) P(b) P(e) + P(j | a) P(m | a) (P(a | b,e) P(b) P(e)
= ???
[Q] Mary calls to say alarm is ringing. what
is the chance that there was an earthquake?
P(X1, …, Xn) =  P(Xi | Parents(Xi))
Bayesian Networks

16. P(X1, …, Xn) =  P(Xi | Parents(Xi))
[Q] P(a) = ???
= P(a | b,e) P(b) P(e) + P(a | b,e) P(b) P(e) +
P(a | b,e) P(b) P(e) + P(a | b,e) P(b) P(e))
= …
[Q] P(m) = ???
P(X1, …, Xn) =  P(Xi | Parents(Xi))
Bayesian Networks

17. [Q] How to implement for queries?2.
[Q] How to implement for queries?
Bayesian Networks

18. How to construct Bayesian networks
1. Choose an ordering of random 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)
i.e., Parents(Xi) are conditionally independent of the others.
P (X1, …, Xn) = Πi =1 P (Xi | X1, …, Xi-1) (by chain rule)
= Πi =1 P (Xi | Parents(Xi)) (by construction)
[Q] Which random variables first? n n
Bayesian Networks

19. How to choose a correct order?2.
How to choose a correct order?
Topics
“Root causes” first,
[Q] What can be a root?
then the variables they directly influence,
and so on,
until the leaves
Can we use causal relations?
For each node, find child nodes (i.e., the child nodes caused by the node);
Example of Weather, Toothache, Catch, and Cavity
Bayesian Networks

20. Topics
Summary
Bayesian networks provide a natural representation for (causally induced) conditional independence.
Topology + CPTs = compact representation of joint distribution
Generally easy for domain experts to construct
Bayesian Networks