1. A Brief Introduction to Bayesian networks
CIS 391 – Introduction to Artificial Intelligence

2. Bayesian networks
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 set of directed edges (link ≈ "directly influences"), yielding a directed, acyclic graph
A conditional distribution for each node given its parents: P (Xi | Parents (Xi))
In the simplest case, conditional distribution represented as a conditional probability table (CPT) giving the distribution over Xi for each combination of parent values
CIS Introduction to AI

3. The Required-By-Law Burglar Example
I'm at work, and my neighbor John calls to say my burglar alarm is ringing, but my neighbor Mary doesn't call. Sometimes it's set off by minor earthquakes. Is there a burglar?
Variables: Burglary, Earthquake, Alarm, JohnCalls, MaryCalls
Network topology reflects "causal" knowledge:
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
CIS Introduction to AI

4. Belief network for example
CIS Introduction to AI

5. Semantics Local semantics give rise to global semantics
Local semantics: given its parents, each node is conditionally independent of everything except its descendants
Global semantics (why?):
CIS Introduction to AI

6. Belief Network Construction Algorithm
Choose an ordering of variables X1, … , Xn
For i = 1 to n
add Xi to the network
select parents from X1, … , Xi-1 such that
CIS Introduction to AI

7. Construction Example Suppose we choose the ordering M, J, A, B, E M J
P(J|M)=P(J)? No
P(A|J,M)=P(A)? P(A|J,M)=P(A|J)? 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|A)? No
P(E|B,A,J,M)=P(E|A,B)?
Yes
CIS Introduction to AI

8. Lessons from the Example
Network less compact: 13 numbers (compared to 10)
Ordering of variables can make a big difference!
Intuitions about causality useful
CIS Introduction to AI

9. Belief Networks and Hidden Variables: An Example
CIS Introduction to AI

10. Compact Conditional Probability Tables
CPT size grows exponentially; continuous variables have infinite CPTs!
Solution: compactly defined canonical distributions
e.g. boolean functions
Gaussian and other standard distributions
CIS Introduction to AI

11. For Inference in Bayesian Networks…
CIS 520 Machine Learning
CIS 520 provides a fundamental introduction to the mathematics, algorithms and practice of machine learning. Topics covered include:
Supervised learning: least squares regression, logistic regression, perceptron, generalized linear models, discriminant analysis, naive Bayes, support vector machines. Model and feature selection, ensemble methods, bagging, boosting. Learning theory: Bias/variance tradeoff. Union and Chernoff/Hoeffding bounds. VC dimension. Online learning.
Unsupervised learning: Clustering. K-means. EM. Mixture of Gaussians. Factor analysis. PCA. MDS. pPCA. Independent components analysis (ICA).
Graphical models: HMMs, Bayesian and Markov networks. Inference. Variable elimination.
Reinforcement learning: MDPs. Bellman equations. Value iteration. Policy iteration. Q-learning. Value function approximation.
(But check prerequisites: CIS 320, statistics, linear algebra)
CIS Introduction to AI