.

Learning Bayesian networks

Slides by Nir Friedman

Learning Bayesian networks

InducerInducerInducerInducerData + Prior information

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.7 .3

.99 .01

.8 .2

be

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BE P(A | E,B)

Known Structure -- Incomplete Data

InducerInducerInducerInducer

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A.9 .1

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.7 .3

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.8 .2

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BE P(A | E,B)

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Network structure is specified Data contains missing values

We consider assignments to missing values

E, B, A<Y,N,N><Y,?,Y><N,N,Y><N,Y,?> . .<?,Y,Y>

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

Maximum Likelihood Estimation in Binomial Data

Applying the MLE principle we get

(Which coincides with what one would expect)

0 0.2 0.4 0.6 0.8 1

L(

:D)

Example:(NH,NT ) = (3,2)

MLE estimate is 3/5 = 0.6

TH

H

NN

N

Learning Parameters for a Bayesian Network

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]1[]1[]1[]1[

MCMAMBME

CABE

D

Training data has the form:

Learning Parameters for a Bayesian Network

E B

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C

Since we assume i.i.d. samples,likelihood function is

m

mCmAmBmEPDL ):][],[],[],[():(

Learning Parameters for a Bayesian Network

E B

A

C

By definition of network, we get

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m

mAmCP

mEmBmAP

mBP

mEP

mCmAmBmEPDL

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MCMAMBME

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Learning Parameters for a Bayesian Network

E B

A

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Rewriting terms, we get

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m

m

m

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mEmBmAP

mBP

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][][][][

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MCMAMBME

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General Bayesian Networks

Generalizing for any Bayesian network:

The likelihood decomposes according to the structure of the network.

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i miii

m iiii

mn

DL

mPamxP

mPamxP

mxmxPDL

):(

):][|][(

):][|][(

):][,],[():( 1 i.i.d. samples

Network factorization

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).

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.

)()(),(),(~

|ii

iiiipax paNpa

paxNpaxii

)(),(ˆ

|i

iipax paN

paxNii

MLE Bayesian (Dirichlet Prior)

Known Structure -- Incomplete Data

InducerInducerInducerInducer

E B

A.9 .1

e

b

e

.7 .3

.99 .01

.8 .2

be

b

b

e

BE P(A | E,B)

? ?

e

b

e

? ?

? ?

? ?

be

b

b

e

BE P(A | E,B) E B

A

Network structure is specified Data contains missing values

We consider assignments to missing values

E, B, A<Y,N,N><Y,?,Y><N,N,Y><N,Y,?> . .<?,Y,Y>

Learning Parameters from Incomplete Data

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

X

Y|X=Hm

X[m]

Y[m]

Y|X=T

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

Gradient Ascent:Follow gradient of likelihood w.r.t. to parameters

L(

|D)

MLE from Incomplete Data Finding MLE parameters: nonlinear optimization problem

L(

|D)

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

MLE from Incomplete Data Finding MLE parameters: nonlinear optimization problem

MLE from Incomplete Data

Both Ideas:Find local maxima only.Require multiple restarts to find approximation to the global maximum.

Gradient Ascent

Main result

Theorem GA:

)],[|,(1)|(log

,,

mopaxPDP

iimpaxpax iiii

Requires computation: P(xi,pai|o[m],) for all i, m

Inference replaces taking derivatives.

Gradient Ascent (cont)

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moPmoP ,

)|][()|][(

1

m paxpax iiii

moPDP

,,

)|][(log)|(log

ii pax

moP

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)|][(

How do we compute ?

Proof:

=1

Gradient Ascent (cont)

Since:

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ii opaxP

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),,','(

ii pax ','

iindi

ndii

d paxPopaPopaxoP

),'|'()|,'(),,','|(

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ii iiiipax pax

ii

pax

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, ','','

)|,,()|(

ipaix

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Gradient Ascent (cont)

Putting all together we get

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moP

moP

DP

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1)|(log

m pax

ii

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1

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mopaxP

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)],[|,(

)],[|,(1)|(log

,,

mopaxPDP

iimpaxpax iiii

Expectation Maximization (EM) A general purpose method for learning from incomplete dataIntuition: 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

Expectation Maximization (EM)

1.30.41.71.6

X Z N (X,Y )X Y #H

THHT

Y

??HTT

TT?TH

HTHT

HHTT

P(Y=H|X=T, Z=T, ) = 0.4

Expected CountsP(Y=H|X=H,Z=T,) = 0.3

Data

Current model

These numbers are placed for illustration; they have not been computed.

X

YZ

EM (cont.)

TrainingData

X1 X2 X3

H

Y1 Y2 Y3

Initial network (G,0)

Expected CountsN(X1)N(X2)N(X3)N(H, X1, X1, X3)N(Y1, H)N(Y2, H)N(Y3, H)

Computation

(E-Step)

Reparameterize

X1 X2 X3

H

Y1 Y2 Y3

Updated network (G,1)

(M-Step)

Reiterate

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.

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.

1. Specify the formula for a pedigree with two parents and two children.

2. Extend the formula for arbitrary pedigrees.

Note that is the same in many local probability tables.