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Parameter

Estimation

Peter NRobinson

EstimatingParametersfrom Data

MaximumLikelihood(ML)Estimation

Betadistribution

Maximum a

posteriori(MAP)Estimation

MAQ

Parameter Estimation

ML vs. MAP

Peter N Robinson

December 14, 2012
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Parameter

Estimation

Peter NRobinson



Betadistribution

Maximum a

posteriori(MAP)Estimation

MAQ

Estimating Parameters from Data

In many situations in bioinformatics, we want to estimate op-timal parameters from data. In the examples we have seen inthe lectures on variant calling, these parameters might be theerror rate for reads, the proportion of a certain genotype, theproportion of nonreference bases etc. However, the hello worldexample for this sort of thing is the coin toss, so we will start

with that.
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Parameter

Estimation

Peter NRobinson



Betadistribution

Maximum aposteriori(MAP)Estimation

MAQ

Coin toss

Lets say we have two coins that are each tossed 10 times

Coin 1: H,T,T,H,H,H,T,H,T,TCoin 2: T,T,T,H,T,T,T,H,T,T

Intuitively, we might guess that coin one is a fair coin, i.e.,P(X =H) = 0.5, and that coin 2 is biased, i.e.,P

(X

=H

)= 0.5
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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

Discrete Random Variable

Let us begin to formalize this. We model the coin toss processas follows.

The outcome of a single coin toss is a random variable Xthat can take on values in a set X ={x

1, x

2, . . . , xn}

In our example, of course, n= 2, and the values arex1 = 0 (tails) and x2 = 1 (heads)

We then have a probability mass function p:X [0, 1];the law of total probability states that xXp(xi) = 1This is a Bernoulli distribution with parameter :

p(X = 1; ) = (1)
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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

Probability of sequence of events

In general, for a sequence of two events X1 and X2, the jointprobability isP(X1, X2) =p(X2|X1)p(X1) (2)

Since we assume that the sequence is iid (identically andindependently distributed), by definition p(X2|X1) =P(X2).Thus, for a sequence ofn events (coin tosses), we have

p(x1, x2, . . . , xn; ) =n

i=1

p(xi; ) (3)

if the probability of heads is 30%, the the probability of thesequence for coin 2 can be calculated as

p(T,T,T,H,T,T,T,H,T,T; ) =2(1 )8 =

3

10

2 710

8(4)
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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

Probability of sequence of events

Thus far, we have considered p(x; ) as a function of x,parametrized by . If we view p(x; ) as a function of, thenit is called the likelihood function.

Maximum likelihood estimation basically chooses a value of

that maximizes the likelihood function given the observed data.
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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

Maximum likelihood for Bernoulli

The likelihood for a sequence of i.i.d. Bernoulli randomvariablesX= [x1, x2, . . . , xn] with xi {0, 1} is then

p(X; ) =n

i=1

p(xi; ) =

n

i=1

xi(1 )1xi (5)

We usually maximize the log likelihood function rather than theoriginal function

Often easier to take the derivative

the log function is monotonically increasing, thus, themaximum (argmax) is the same

Avoid numerical problems involved with multiplying lotsof small numbers
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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

Log likelihood

Thus, instead of maximizing this

p(X; ) =n

i=1

xi(1 )1xi (6)

we maximize this

log p(X; ) = logni=1

xi(1 )1xi

=

n

i=1 log xi(1 )1xi

=ni=1

log xi + log(1 )1xi

=n

i=1

[xilog + (1 xi) log(1 )]
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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

Log likelihood

Note that one often denotes the log likelihood function withthe symbol L= log p(X; ).

A function f defined on a subset of the real numbers with real

values is called monotonic (also monotonically increasing, in-creasing or non-decreasing), if for all x and y such that xyone has f(x)f(y)

Thus, the monotonicity of the log function guarantees that

argmax

p(X; ) = argmax

log p(X; ) (7)
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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

ML estimate

The ML estimate of the parameter is then

argmax

ni=1

[xilog + (1 xi) log(1 )] (8)

We can calculate the argmax by setting the first derivative

equal to zero and solving for

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Parameter

Estimation

Peter NRobinson



Betadistribution


MAQ

ML estimate

Thus

log p(X; ) =n

i=1

[xilog + (1 xi) log(1 )]

=n

i=1

xi

log +

ni=1

(1 xi)

log(1 )

= 1

ni=1

xi 1

1

ni=1

(1 xi)

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Parameter

Estimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

ML estimate

and finally, to find the maximum we set

log p(X; ) = 0:

0 = 1

ni=1

xi 1

1

ni=1

(1 xi)

1

=n

i=1

(1 xi)ni=1xi

1

1 =

ni=11

ni=1xi

1

1

=

nni=1xi

ML = 1

n

ni=1

xi

Reassuringly, the maximum likelihood estimate is just theproportion of flips that came out heads.

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Parameter

Estimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

Problems with ML estimation

Does it really make sense that

H,T,H,T = 0.5H,T,T,T = 0.25

T,T,T,T = 0.0

ML estimation does not incorporate any prior knowledge and

does not generate an estimate of the certainty of its results.

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Parameter

Estimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

Maximum a posteriori Estimation

Bayesian approaches try to reflect our belief about . In thiscase, we will consider to be a random variable.

p(|X) =

p(X|)p()

p(X) (9)

Thus, Bayes law converts our prior belief about the parameter (before seeing data) into a posterior probability, p(|X), byusing the likelihood function p(X|). The maximum

a-posteriori (MAP) estimate is defined as

MAP= argmax

p(|X) (10)
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Parameter

Estimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

Maximum a posteriori Estimation

Note that because p(X) does not depend on , we have

MAP

= argmax

p(|X)

= argmax

p(X|)p()

p(X)

= argmax

p(X|)p()

This is essentially the basic idea of the MAP equation used bySNVMix for variant calling
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Parameter

Estimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

MAP Estimation; What does it buy us?

To take a simple example of a situation in which MAP estima-tion might produce better results than ML estimation, let usconsider a statistician who wants to predict the outcome of thenext election in the USA.

The statistician is able to gather data on party preferencesby asking people he meets at the Wall Street Golf Club1

which party they plan on voting for in the next election

The statistician asks 100 people, seven of whom answerDemocrats. This can be modeled as a series ofBernoullis, just like the coin tosses.

In this case, the maximum likelihood estimate of theproportion of voters in the USA who will vote democraticis ML= 0.07.

1i.e., a notorious haven of ultraconservativeRepublicans

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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

MAP Estimation; What does it buy us?

Somehow, the estimate of ML= 0.07 doesnt seem quite rightgiven our previous experience that about half of the electoratevotes democratic, and half votes republican. But how shouldthe statistician incorporate this prior knowledge into hisprediction for the next election?

The MAP estimation procedure allows us to inject our priorbeliefs about parameter values into the new estimate.

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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

Beta distribution: Background

The Beta distribution is appropriate to express prior belief abouta Bernoulli distribution. The Beta distribution is a family ofcontinuous distributions defined on [0, 1] and parametrized bytwo positive shape parameters, and

p() = 1

B(, )1 (1 )1

here, [0, 1], and

B(, ) = (+)

() ()

where is the Gamma function(extension of factorial). 0.0 0.2 0.4 0.6 0.8 1.0

0.

0

0.

5

1.

0

1.

5

2.

0

2.5

beta distribution

x

p

df

==0.5=5, =1=1, =3==2=2, =5
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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Random variables are either discrete (i.e., they can assume oneof a list of values, like the Bernoulli with heads/tails) or con-tinuous (i.e., they can take on any numerical value in a certaininterval, like the Beta distribution with ).

A probability density function (pdf) of a continuousrandom variable, is a function that describes the relativelikelihood for this random variable to take on a givenvalue, i.e., p() : R R+ such that

Pr( (a, b)) =

ba

p()d (11)

The probability that the value of lies between a and bis given by integrating the pdf over this region
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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Recall the difference between a PDF (for continuous randomvariable) and a probability mass function (PMF) for a discreterandom variable

A PMF is defined asPr

(X =xi) =pi, with 0 pi1and

ipi= 1

e.g., for a fair coin toss (Bernoulli),Pr(X = heads) = Pr(X = tails) = 0.5

In contrast, for a PDF, there is no requirement thatp() 1, but we do have +

p()d= 1 (12)
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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


We calculate the PDF for the Beta distribution for asequence of values 0, 0.01, 0.02, . . . , 1.00 in R as follows

x

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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Themodeof a continuous probability distribution is thevalue x at which its probability density function has itsmaximum value

The mode of the Beta(, ) distribution has itsmodeat

1+ 2 (13)

alpha

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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


The code from the previous slide leads to

0.0 0.2 0.4 0.6 0.8 1.0

0.

0

0.

5

1.

0

1.

5

2.0

2.

5

=7 =3

x

pdf

The mode, shown as the dotted red line, is calculated as 1

+ 2=

7 1

7 + 3 2= 0.75
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Parameter

EstimationPeter N

Robinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

Maximum a posteriori (MAP) Estimation

With all of this information in hand, lets get back to MAPestimation!

Going back to Bayes rule, again, we seek the value ofthat maximizes the posterior Pr(|X):

Pr(|X) = Pr(X|)Pr()

Pr(X) (14)
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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


We then have

MAP = argmax

Pr(|X)

= argmax

Pr(X|)Pr()

Pr(X)

= argmax

Pr(X|)Pr()

= argmaxxiX

Pr(xi|)Pr()

( )
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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


As we saw above for maximum likelihood estimation, it is easierto calculate the argmax for the logarithm

argmax

Pr(|X) = argmax

log Pr(|X)

= argmax

logxiX

Pr(xi|) Pr()

= argmax

xiX

{log Pr(xi|)} + log Pr()

M i i i (MAP) E i i
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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Lets go back now to our problem of predicting the results ofthe next election. Essentially, we plug in the equations for thedistributions of the likelihood (a Bernoulli distribution) and theprior (A Beta distribution).

Pr(|X) Pr(xi|) Pr()

posterior

Likelihood (Bernoulli)

prior (Beta)

M i i i (MAP) E i i
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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Furthermore

L = log Pr(|X)

= logi

Bernoulli(xi|) Beta(|, )=

i

log Bernoulli(xi|) + log Beta(|, )

We solve for MAP= argmax Las follows

argmax

i

log Bernoulli(xi|) + log Beta(|, )

Note that this is almost the same as the ML estimate except

that we now have an additional term resultingfromtheprior

M i t i i (MAP) E ti ti
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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Again, we find the maximum value of by setting the firstderivative ofLequal to zero and solving for

L =i

log Bernoulli(xi|) +

log Beta(|, )

The first term is the same as for ML2, i.e.

i

log Bernoulli(xi|) = 1

ni=1

xi 1

1

ni=1

(1xi) (16)

2see slide 11

Ma im m a osteriori (MAP) Estimation
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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


To find the second term, we note

log Beta(|, ) =

log

( + )

()() 1 (1 )1

=

log ( + )

()()+

log 1

(1 )1

= 0 +

log 1 (1 )1

=

(1)

log + (1)

(1 )

= 1

11

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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


To find MAP, we now set

L= 0 and solve for

0 =

L

= 1

n

i=1xi

1

1

n

i=1(1 xi) +

1

1

1

and thus

ni=1

(1 xi) + 1

= (1 )

i

xi+ 1

n

i=1

(1 xi) +i

xi+ 1 + 1

=i

xi+ 1

ni=1

1 + + 2

=

i

xi+ 1

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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Finally, if we let our Bernoulli distribution be coded as

Republican=1 and Democrat=0, we have that

ixi =nr where nrdenotes the number of Republican voters

(17)Then,

ni=1

1 + + 2

=

i

xi+ 1

[n+ + 2] = nR+ 1

and finally

MAP= nR+ 1

n++ 2

(18)

And now our prediction

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ParameterEstimation

Peter NRobinson



Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


It is useful to compare the ML and the MAP predictions. Noteagain that and are essentially the same thing as pseudo-counts, and the higher their value, the more the prior affectsthe final prediction (i.e., the posterior).

Recall that in our poll of 100 members of the Wall Street Golfclub, only seven said they would vote democratic. Thus

n= 100

nr = 93

We will assume that the mode of our prior belief is that50% of the voters will vote democratic, and 50%republican. Thus, = . However, different values foralpha and beta express different strengths of prior belief

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ParameterEstimation

Peter NRobinson


Maximum

Likelihood(ML)Estimation

Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


n nR ML MAP

100 93 1 1 0.93 0.93100 93 5 5 0.93 0.90100 93 100 100 0.93 0.64100 93 1000 1000 0.93 0.52100 93 10000 10000 0.93 0.502

Thus, MAP pulls the estimate towards the prior to an extent that depends on the strength of the prior

The use of MAP in MAQ
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ParameterEstimation

Peter NRobinson


Maximum


Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

The use of MAP in MAQ

Recall from the lecture that we call the posterior probabilitiesof the three genotypes given the data D, that is a column withn aligned nucleotides and quality scores of which k correspondto the reference a and n k to a variant nucleotide b.

p(G =a, a|D) p(D|G =a, a)p(G =a, a)

n,k (1 r)/2

p(G =b, b|D) p(D|G =b, b)p(G=b, b)

n,nk (1 r)/2p(G=a, b|D) p(D|G =a, b)p(G=a, b)

n

k

1

2n r

MAQ: Consensus Genotype Calling
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ParameterEstimation

Peter NRobinson


Maximum


Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


Note that MAQ does not attempt to learn the parameters,rather it uses user-supplied parameter r which roughly corre-sponds to in the election.

MAQ calls the the genotype with the highest posteriorprobability:

g= argmaxg(a,a,a,b,b,b)

p(g|D)

The probability of this genotype is used as a measure ofconfidence in the call.

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ParameterEstimation

Peter NRobinson


Maximum


Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


A major problem in SNV calling is false positive heterozygous

variants. It seems less probable to observe a heterozygous callat a position with a common SNP in the population. For thisreason, MAQ uses a different prior (r) for previously knownSNPs (r= 0.2) and new SNPs (r= 0.001).

Let us examine the effect of these two priors on variant calling.In R, we can write

p(G =a, b|D)

n

k1

2n r

as

> dbinom(k,n,0.5) * r

where kis the number of non-reference bases, n is the total number of bases, and 0.5 corresponds to the

probability of seeing a non-ref base given that the true genotypeisheterozygous.

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ParameterEstimation

Peter NRobinson


Maximum


Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


A major problem in SNV calling is false positive heterozygous

variants. It seems less probable to observe a heterozygous callat a position with a common SNP in the population. For thisreason, MAQ uses a different prior (r) for previously knownSNPs (r= 0.2) and new SNPs (r= 0.001).

Let us examine the effect of these two priors on variant calling.In R, we can write

p(G =a, b|D)

n

k1

2n r

as

> dbinom(k,n,0.5) * r

where kis the number of non-reference bases, n is the total number of bases, and 0.5 corresponds to the

probability of seeing a non-ref base given that the true genotypeisheterozygous.

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ParameterEstimation

Peter NRobinson


Maximum


Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ


0 5 10 15 20

.

.

.

r=0.001

k

A

0 5 10 15 20

0.0

0.

4

0.

8

r=0.2

k

B

The figures show the posterior for the heterozygousgenotype according to the simplified MAQ algorithmdiscussed in the previous lecture

The prior r= 0.0001 means than positions with 5 or lessALT bases do not get called as heterozygous, whereas theprior with r= 0.2 means that positions with 5 bases do

get a het call


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ParameterEstimation

Peter NRobinson


Maximum


Betadistribution

Maximum aposteriori

(MAP)Estimation

MAQ

Q yp g

What have we learned?

Get used to Bayesian techniques important in many areasof bioinformatics

understand the difference between ML and MAPestimation

understand the Beta function, priors, pseudocounts

Note that MAQ is no longer a state of the art algorithm,

and its use of the MAP framework is relatively simplisticNonetheless, a good introduction to this topic, and we willsee how these concepts are used in EM today
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mlvsmap

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