1. CS498-EA Reasoning in AI Lecture #20
Instructor: Eyal Amir
Fall Semester 2009
Who is in this class?
Some slides in this set were adopted from Eran Segal

2. Summary of last time: Inference
We presented the variable elimination algorithm
Specifically, VE for finding marginal P(Xi) over one variable, Xi from X1,…,Xn
Order on variables such that
One variable Xj eliminated at a time
(a) Move unneeded terms (those not involving Xj) outside summation over Xj
(b) Create a new potential function, fXj(.) over other variables appearing in the terms of the summation at (a)
Works for both BNs and MFs (Markov Fields)

3. Exact Inference Treewidth methods: Variable elimination
Clique tree algorithm
Treewidth

4. Today: Learning in Graphical Models
Parameter Estimation
Maximum Likelihood
Complete Observations
Naïve Bayes
Not so Naïve Bayes

5. Learning Introduction
So far, we assumed that the networks were given
Where do the networks come from?
Knowledge engineering with aid of experts
Automated construction of networks
Learn by examples or instances 5

6. maximize i P(bi,ci,ei; w0, w1)
Parameter estimation
Maximum likelihood estimation:
maximize i P(bi,ci,ei; w0, w1)

7. Learning Introduction
Input: dataset of instances D={d[1],...d[m]}
Output: Bayesian network
Measures of success
How close is the learned network to the original distribution
Use distance measures between distributions
Often hard because we do not have the true underlying distribution
Instead, evaluate performance by how well the network predicts new unseen examples (“test data”)
Classification accuracy
How close is the structure of the network to the true one?
Use distance metric between structures
Hard because we do not know the true structure
Instead, ask whether independencies learned hold in test data 7

8. Prior Knowledge Prespecified structure Prespecified variables
Learn only CPDs
Prespecified variables
Learn network structure and CPDs
Hidden variables
Learn hidden variables, structure, and CPDs
Complete/incomplete data
Missing data
Unobserved variables 8

9. Learning Bayesian NetworksX1 X2
Inducer
Data
Prior information Y
P(Y|X1,X2) X1 X2 y0 y1
x10
x20 1
x21
0.2
0.8
x11
0.1
0.9
0.02
0.98 9

10. Known Structure, Complete Data
Goal: Parameter estimation
Data does not contain missing values X1 X2 X1 X2
Inducer
Initial network Y Y X1 X2 Y
x10
x21 y0
x11
x20 y1
P(Y|X1,X2) X1 X2 y0 y1
x10
x20 1
x21
0.2
0.8
x11
0.1
0.9
0.02
0.98
Input Data 10

11. Unknown Structure, Complete Data
Goal: Structure learning & parameter estimation
Data does not contain missing values X1 X2 X1 X2
Inducer
Initial network Y Y X1 X2 Y
x10
x21 y0
x11
x20 y1
P(Y|X1,X2) X1 X2 y0 y1
x10
x20 1
x21
0.2
0.8
x11
0.1
0.9
0.02
0.98
Input Data 11

12. Known Structure, Incomplete Data
Goal: Parameter estimation
Data contains missing values X1 X2 X1 X2
Inducer
Initial network Y Y X1 X2 Y ?
x21 y0
x11
x10
x20 y1
P(Y|X1,X2) X1 X2 y0 y1
x10
x20 1
x21
0.2
0.8
x11
0.1
0.9
0.02
0.98
Input Data 12

13. Unknown Structure, Incomplete Data
Goal: Structure learning & parameter estimation
Data contains missing values X1 X2 X1 X2
Inducer
Initial network Y Y X1 X2 Y ?
x21 y0
x11
x10
x20 y1
P(Y|X1,X2) X1 X2 y0 y1
x10
x20 1
x21
0.2
0.8
x11
0.1
0.9
0.02
0.98
Input Data 13

14. Parameter Estimation Input Goal: Learn CPD parameters
Network structure
Choice of parametric family for each CPD P(Xi|Pa(Xi))
Goal: Learn CPD parameters
Two main approaches
Maximum likelihood estimation
Bayesian approaches 14

15. Biased Coin Toss Example
Coin can land in two positions: Head or Tail
Estimation task
Given toss examples x[1],...x[m] estimate P(H)= and P(T)=1-
Assumption: i.i.d samples
Tosses are controlled by an (unknown) parameter 
Tosses are sampled from the same distribution
Tosses are independent of each other 15

16. Biased Coin Toss Example
Goal: find [0,1] that predicts the data well
“Predicts the data well” = likelihood of the data given 
Example: probability of sequence H,T,T,H,H
0.2
0.4
0.6
0.8 1 
L(D:) 16

17. Maximum Likelihood Estimator
Parameter  that maximizes L(D:)
In our example, =0.6 maximizes the sequence H,T,T,H,H
0.2
0.4
0.6
0.8 1 
L(D:) 17

18. Maximum Likelihood Estimator
General case
Observations: MH heads and MT tails
Find  maximizing likelihood
Equivalent to maximizing log-likelihood
Differentiating the log-likelihood and solving for  we get that the maximum likelihood parameter is: 18

19. Sufficient Statistics
For computing the parameter  of the coin toss example, we only needed MH and MT since
 MH and MT are sufficient statistics 19

20. Sufficient Statistics
Definition: Function s(D) is a sufficient statistics from instances to a vector in k if for any two datasets D and D’ and any  we have
Datasets
Statistics 20

21. Sufficient Statistics for Multinomial
A sufficient statistics for a dataset D over a variable Y with k values is the tuple of counts <M1,...Mk> such that Mi is the number of times that the Y=yi in D
Sufficient statistic
Define s(x[i]) as a tuple of dimension k
s(x[i])=(0,...0,1,0,...,0)
(1,...,i-1)
(i+1,...,k) 21

22. Bias and Variance of Estimator

23. Next Time