1. Learning Bayesian networks
Slides by Nir Friedman .

2. Learning Bayesian networks
Inducer E R B A C
Data +
Prior information .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B)

3. Known Structure -- Incomplete Data
E, B, A
<Y,N,N>
<Y,?,Y>
<N,N,Y>
<N,Y,?> .
<?,Y,Y> E B A
Inducer .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B) ? e b B E
P(A | E,B) E B A
Network structure is specified
Data contains missing values
We consider assignments to missing values

4. Known Structure / Complete Data
Given a network structure G
And choice of parametric family for P(Xi|Pai)
Learn parameters for network from complete data
Goal
Construct a network that is “closest” to probability distribution that generated the data

5. Maximum Likelihood Estimation in Binomial Data
Applying the MLE principle we get
(Which coincides with what one would expect)
0.2
0.4
0.6
0.8 1
L( :D)
Example:
(NH,NT ) = (3,2)
MLE estimate is 3/5 = 0.6

6. Learning Parameters for a Bayesian Network
Training data has the form: E B A C

7. Learning Parameters for a Bayesian Network
Since we assume i.i.d. samples, likelihood function is E B A C

8. Learning Parameters for a Bayesian Network
By definition of network, we get E B A C

9. Learning Parameters for a Bayesian Network
Rewriting terms, we get E B A C

10. General Bayesian Networks
Generalizing for any Bayesian network:
The likelihood decomposes according to the structure of the network.
i.i.d. samples
Network factorization

11. General Bayesian Networks (Cont.)
Complete Data  Decomposition
 Independent Estimation Problems
If the parameters for each family are not related, then they can be estimated independently of each other.
(Not true in Genetic Linkage analysis).

12. Learning Parameters: Summary
For multinomial we collect sufficient statistics which are simply the counts N (xi,pai)
Parameter estimation
Bayesian methods also require choice of priors
Both MLE and Bayesian are asymptotically equivalent and consistent.
MLE
Bayesian (Dirichlet Prior)

13. Known Structure -- Incomplete Data
E, B, A
<Y,N,N>
<Y,?,Y>
<N,N,Y>
<N,Y,?> .
<?,Y,Y> E B A
Inducer .9 .1 e b .7 .3
.99
.01 .8 .2 B E
P(A | E,B) ? e b B E
P(A | E,B) E B A
Network structure is specified
Data contains missing values
We consider assignments to missing values

14. Learning Parameters from Incomplete DataX
Y|X=H m
X[m]
Y[m]
Y|X=T
Incomplete data:
Posterior distributions can become interdependent
Consequence:
ML parameters can not be computed separately for each multinomial
Posterior is not a product of independent posteriors

15. Learning Parameters from Incomplete Data (cont.).
In the presence of incomplete data, the likelihood can have multiple global maxima
Example:
We can rename the values of hidden variable H
If H has two values, likelihood has two global maxima
Similarly, local maxima are also replicated
Many hidden variables  a serious problem H Y

16. MLE from Incomplete Data
Finding MLE parameters: nonlinear optimization problem
Gradient Ascent:
Follow gradient of likelihood w.r.t. to parameters
L(|D) 

17. MLE from Incomplete Data
Finding MLE parameters: nonlinear optimization problem
Expectation Maximization (EM):
Use “current point” to construct alternative function (which is “nice”)
Guaranty: maximum of new function is better scoring than the current point
L(|D) 

18. MLE from Incomplete Data
Both Ideas:
Find local maxima only.
Require multiple restarts to find approximation to the global maximum.

19. Gradient Ascent Main result Theorem GA:
Requires computation: P(xi,pai|o[m],) for all i, m
Inference replaces taking derivatives.

20. Gradient Ascent (cont)
Proof: å Q = m pa x i o P D , ) | ] [ (
log q å Q = m pa x i o P , ) | ] [ ( 1 q
How do we compute ?

21. Gradient Ascent (cont)
Since: i pa x o P , ' ) | ( q Q = å i pa x nd d o P , ' ) | ( q Q = å =1 i pa x ' , nd d P o ) | ( q Q = i pa x o P ' , ) ( q Q =

22. Gradient Ascent (cont)
Putting all together we get

23. Expectation Maximization (EM)
A general purpose method for learning from incomplete data
Intuition:
If we had access to counts, then we can estimate parameters
However, missing values do not allow to perform counts
“Complete” counts using current parameter assignment

24. Expectation Maximization (EM)Y Z
Data
Expected Counts
P(Y=H|X=H,Z=T,) = 0.3 X Y Z
N (X,Y )
HTHHT
??HTT
TT?TH X Y #
Current
model
HTHT
HHTT
These numbers are placed for illustration; they have not been computed.
P(Y=H|X=T, Z=T, ) = 0.4

25. EM (cont.)  Training Data Reiterate Expected Counts
Initial network (G,0)
Reparameterize X1 X2 X3 H Y1 Y2 Y3
Updated network (G,1)
(M-Step)
Expected Counts
N(X1)
N(X2)
N(X3)
N(H, X1, X1, X3)
N(Y1, H)
N(Y2, H)
N(Y3, H)
Computation
(E-Step) X1 X2 X3 H Y1 Y2 Y3 
Training
Data

26. Expectation Maximization (EM)
In practice, EM converges rather quickly at start but converges slowly near the (possibly-local) maximum.
Hence, often EM is used few iterations and then Gradient Ascent steps are applied.

27. Final Homework
Question 1: Develop an algorithm that given a pedigree input, provides the most probably haplotype of each individual in the pedigree. Use the Bayesian network model of superlink to formulate the problem exactly as a query. Specify the algorithm at length discussing as many details as you can. Analyze its efficiency. Devote time to illuminating notation and presentation.
Question 2: Specialize the formula given in Theorem GA for  in genetic linkage analysis. In particular, assume exactly 3 loci: Marker 1, Disease 2, Marker 3, with  being the recombination between loci 2 and 1 and 0.1-  being the recombination between loci 3 and 2.
Specify the formula for a pedigree with two parents and two children.
Extend the formula for arbitrary pedigrees.
Note that  is the same in many local probability tables.