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Learning – EM in The ABO locusTutorial #9

© Ilan Gronau.

Based on original slides of Ydo Wexler & Dan Geiger

2

Genotype statistics

Mendelian Genetics:• locus - a particular location on a chromosome (genome)

- Each locus has two copies – alleles (one paternal and one maternal)- Each copy has several relevant states - genotypes

• locus genotype is determined by the combined genotype of both copies.• locus genotype yields phenotype (physical features)

NN tsts ,,

We wish to estimate the distribution of all possible genotypes.

Suppose we randomly sample N individuals and found the

number Ns,t.

The MLE is given by: Sampling genotypes is costlySampling phenotypes is cheap

3

The ABO locus

• ABO locus determines blood-type

• It has six possible genotypes {a/a, a/o, b/o, b/b, a/b, o/o}.

• They lead to four possible phenotypes: {A, B, AB, O}

We wish to estimate the proportion in a population of the 6

genotypes.

- Sample genotype – sequence a genomic region

- Sample phenotype - checking presence of antibodies (simple

blood test)

Problem: phenotype doesn’t reveal genotype (in case of

A,B)

4

The ABO locus

Problem: phenotype doesn’t reveal genotype

The probabilistic model: Allele genotypes are distributed

i.i.d w.p a ,b ,o, and determine probabilities for locus genotypes:

• a/b=2a b ; a/o=2a o ; b/o=2b o

• a/a= a2 ; b/b=b

2 ; o/o=o2

This implies probabilities for phenotypes:

• Pr[P=A |Θ] = a/a+a/o = a2+2a o

• Pr[P=B |Θ] = b/b+b/o = b2+2b o

• Pr[P=AB |Θ] = a/b= 2a b

• Pr[P=O |Θ] = o/o = o2

Hardy-Weinbergequilibrium

Θ - model parameter set

Θ={a ,b ,o}

5

Likelihood of phenotype data

Given a population phenotype sample: Data =

{B,A,B,B,O,A,B,A,O,B, AB}

the likelihood of our parameter set Θ={a ,b ,o} is:

3 5 212 2 2Pr[ | ] 2 2 2a a o b b o a b oData A B AB O

• Maximum of this function yields the MLE

Use EM to obtain this

6

The EM algorithm

The setting for the algorithm:

• Our data is a series of outcomes of experiments.

• Each experiment is conducted identically and independently.

• The outcome of an experiment is a function of values selected

for a set of discrete random variables – X1,..Xn .

• The actual values selected for X1,..Xn may be hidden from us.

We wish to find the MLE of the p.d’s for X1,..Xn .

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The EM algorithmThe setting for the algorithm:• Our data is a series of outcomes of experiments.• Each experiment is conducted identically and independently.• The outcome of an experiment is a function of values selected for a set

of discrete random variables – X1,..Xn .

• The actual values selected for X1,..Xn may be hidden from us.

We wish to find the MLE of the p.d’s for X1,..Xn .Examples:

1.Genotyping in the ABO locus:• Single hidden variable X – a single allele genotype (a,b, or o)

• Model parameters - Θ={a ,b ,o}

2.Hidden Markov Models:

• Two hidden variables Ts , Es for every state state s

(Es – chooses signal ; Ts – chooses next state)

• Model parameters – transition and emmission probabilities.

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The EM algorithm

Start with some set of parameters- Θ.

Iterate until convergence:

• E-step:

calculate the expected count for every possible result of every hidden variable in the model, as implied by data and Θ

• M-step:

For every hidden variable:

- Use expected counts as statistics to yield Θ’ MLE(data,Θ)

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The EM algorithmE-step:

calculate the expected count for every possible result of every hidden variable in the model, as implied by data and Θ

M-step:

For every hidden variable:- Use expected counts as statistics to yield Θ’ MLE(data,Θ)

In our example:

• Single hidden variable X – a single allele genotype (a,b, or o)

• Model parameters - Θ={a ,b ,o}

E-step: count the expected number of a,b,o alleles in

population(total number of counts - 2n).

M-step: set ’a = #a/2n ; ’b = #b/2n ; ’o = #o/2n .

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E-step calculations – gene counting

genotype

a/o

a/a

b/o

b/b

a/b

o/o

gene count

a b o

1 0 1

2 0 0

0 1 1

0 2 0

1 1 0

0 0 2

pheno-

type

A

B

AB

O

prob

2 a o 2

a

2 b o 2

b

2 a b 2

o

gene count

a b o

0

0

1 1 0

0 0 2

2

2o

o a

2a

o a

2

2o

o a

2

2o

o b

2

2o

o b

2b

o b

1*

+2*

1*

1*

+2*

1*

observed outcome

of “experiment”

result(s) ofhidden variables

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Datatype #people

A 100

B 200

AB 50

O 50

We start with an initial guess: 0 = {0.2, 0.2, 0.6}

A numeric example

Sufficient statistics:

nA , nB , nAB , nO

a b o

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1st iteration: 0= {0.2, 0.2, 0.6}

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2( )100 200 0 50 1 50 0 164

(2 )o a

o a

A numeric example - execution of EMData

type #people

A 100

B 200

AB 50

O 50E-step: A B AB O

E[(#a)] =

E[(#b)] =

E[(#o)] =

47

2( )100 0 200 50 1 50 0 278

(2 )o b

o b

17

2 2100 200 50 0 50 2 357

(2 ) (2 )o o

o a o b

800 = 2nM-step:2 4 1

7 7 7164 278 357' 0.205 ; ' 0.348 ; ' 0.447

800 800 800a b o

1= {0.205, 0.348, 0.447}

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A numeric example - execution of EMData

type #people

A 100

B 200

AB 50

O 50E-step: A B AB O

E[(#a)] =

E[(#b)] =

E[(#o)] =

800 = 2nM-step:

2nd iteration: 1= {0.205, 0.348, 0.447}

2= {0.211, 0.383, 0.406}

168.66 306.04 325.3' 0.211 ; ' 0.383 ; ' 0.406

800 800 800a b o

2( )100 200 0 50 1 50 0 168.66

(2 )o a

o a

2( )100 0 200 50 1 50 0 306.04

(2 )o b

o b

2 2

100 200 50 0 50 2 325.3(2 ) (2 )

o o

o a o b

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E-step:

2( )[# ] 1

(2 )

2( )[# ] 1

(2 )

2 2[# ] 2

(2 ) (2 )

o aA AB

o a

o bB AB

o b

o oA B O

o a o b

E a n n

E b n n

E o n n n

Sufficient statistics – nA , nB , nAB , nO

M-step: [# ] [# ] [# ]; ;

2 2 2a b b

E a E b E b

n n n

EM algorithm for the ABO locus - summary

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Iteration update formula:

2( )11

2 (2 )

2( )11

2 (2 )

2 212

2 (2 ) (2 )

o aa A AB

o a

o bb B AB

o b

o oo A B O

o a o b

n nn

n nn

n n nn

Sufficient statistics – nA , nB , nAB , nO ,

EM algorithm for the ABO locus - summary

16

EM algorithm – ABO exampleData

type #people

A 100

B 200

AB 50

O 50

0.20

0.38

0.42

a,

b, o

Learning iteration

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EM algorithm – ABO exampleData

type #people

A 100

B 200

AB 50

O 50

0.20

0.38

0.42

a,

b, o

Learning iteration

good convergence(maybe)

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Alternative solution

Alternative view:

• Single hidden variable X’ – a maternal allele genotype (a,b, or

o)

• Model parameters - Θ={a ,b ,o}

E-step: count the expected number of maternal a,b,o alleles

in population (total number of counts - n).

M-step: set ’a = #a/n ; ’b = #b/n ; ’o = #o/n .

Initial view:• Single hidden variable X – a single allele genotype (a,b, or o)

• Model parameters - Θ={a ,b ,o}

E-step: count the expected number of a,b,o alleles in population(total number of counts - 2n).

M-step: set ’a = #a/2n ; ’b = #b/2n ; ’o = #o/2n .

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count

a b o

1* 0

1*

0 1*

1*

1/2 1/2 0

0 0 1

E-step calculations – gene countingmat.gen.

o

a

o

b

a

b

o

count

a b o

0 0 1

1 0 0

0 0 1

0 1 0

1 0 0

0 1 0

0 0 1

pheno-

type

A

B

AB

O

prob

2a o

o a

2

o

o a

2o

o b

observed outcome

of “experiment”

result(s) ofhidden variables

b o

( )b b o

a b

2o

a b

a o

( )a a o

2b o

o b

Exactly ½ of what we got by gene counting

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Iteration update formula:

( )1 1

(2 ) 2

( )1 1

(2 ) 2

11

(2 ) (2 )

o aa A AB

o a

o bb B AB

o b

o oo A B O

o a o b

n nn

n nn

n n nn

Sufficient statistics – nA , nB , nAB , nO ,

EM algorithm for the ABO locus - summary