1. Bayesian Networks
Probability In AI

2. Bayes’ Theorem
P(A|B) = P(B|A) P(A) / P(B)

3. Example Bowl 1 – 10 red and 30 white balls
Randomly pick a bowl and then a ball from it
What is P(Bowl 1 | white ball) ?
By Bayes’
P(white|bowl 1) P(bowl 1) / P(white)
P(white) = .5 P(white|bowl 1) + .5 P(white|bowl 2)

4. Bayesian Network G=(V,E) be a DAG Let each node be a random variable
For node a the nodes connected to a are {bi}
P(a|{bi}) (written in Conditional Probability Tables)

5. Example (From Russell and Novrig Chapt 14)
Burglary
Eartthquake
P(B)=.001
P(E)=.002
B E P(A)
T T
T F
F T
F F
Alarm
JohnCalls
MaryCalls
A P(M) T F
A P(J) T F

6. Semantics The graph represents the full joint probability distribution
P(x1, …. xn) is the probability of the set of assignments to the variables
P(x1, …. xn) = Pi=1,n P(xi | parents(Xi))

7. Construction Nodes from the set of random variables Links
Order them with causes before effects(not required, but simpler!)
Links
For each node determine the set of parents
Link them
Define Conditional Probablity Table

8. Query
Given an observed event, what is posterior probability for the query variables.
Ex: P(Burglary | JohnCalles = true , MaryCalles = true)
Answer is <0.284,0.716>

9. Method Single Variable by enumeration
Find P(X,e)
X is query variable
e is event : on the evidence variables
Y is a set of hidden variables
P(X|e) = a S P(X,e,y)
The constant a is to normalize

10. P(Burglary | JohnCalles = true , MaryCalles = true)
P(B| j,m) =a P(B,j,m) = a Se Sa P(b,j,m,e,a)
For CPT entries:
For burglary = true:
= a Se Sa P(b)P(e)P(a|b,e)P(j|a)P(m|a)
This is
For the false case
The a is computed to give a sum = 1

11. Variable Elimination
A query method that works by removing the unobserved variables
More efficient

12. Summary