Download Rich Probabilistic Models for Genomic Data

Maximum Likelihood Estimation
& Expectation Maximization
Lectures 3 – Oct 5, 2011
CSE 527 Computational Biology, Fall 2011
Instructor: Su-In Lee
TA: Christopher Miles
Monday & Wednesday 12:00-1:20
Johnson Hall (JHN) 022
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Outline
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Probabilistic models in biology
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Model selection problem
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Mathematical foundations
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Bayesian networks
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Learning from data
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Maximum likelihood estimation
Maximum a posteriori (MAP)
Expectation and maximization
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Parameter Estimation
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Assumptions

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For example,
{i0,d1,g1,l0,s0}
Fixed network structure
Fully observed instances of the network variables: D={d[1],…,d[M]}
Maximum likelihood estimation (MLE)!
“Parameters” of the
Bayesian network
from Koller & Friedman
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The Thumbtack example
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Parameter learning for a single variable.
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Variable

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X: an outcome of a thumbtack toss
Val(X) = {head, tail}
X
Data
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A set of thumbtack tosses: x[1],…x[M]
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Maximum likelihood estimation
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Say that P(x=head) = Θ, P(x=tail) = 1-Θ
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Definition: The likelihood function
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P(HHTTHHH…<Mh heads, Mt tails>; Θ) =
L(Θ : D) = P(D; Θ)
Maximum likelihood estimation (MLE)
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Given data D=HHTTHHH…<Mh heads, Mt tails>, find Θ
that maximizes the likelihood function L(Θ : D).
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Likelihood function
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MLE for the Thumbtack problem

Given data D=HHTTHHH…<Mh heads, Mt tails>,
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MLE solution Θ* = Mh / (Mh+Mt ).
Proof:
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Continuous space

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Assuming sample x1, x2,…, xn is from a
parametric distribution f (x|Θ) , estimate Θ.
Say that the n samples are from a normal
distribution with mean μ and variance σ2.
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Continuous space (cont.)

Let Θ1=μ, Θ2= σ2
L(1 , 2 : x1 , x2 ,..., xn ) 
log L(1 , 2 : x1 , x2 ,..., xn ) 

log L(1 , 2 : x1 , x2 ,..., xn ) 
1

log L(1 , 2 | x1 , x2 ,..., xn ) 
 2
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Any Drawback?

Is it biased?
Is it? Yes. As an extreme, when n = 1, ˆ2 =0.
ˆ
 The MLE  2 systematically underestimates θ2 .
Why? A bit harder to see, but think about n = 2. Then
θ1 is exactly between the two sample points, the
position that exactly minimizes the expression
for .
ˆ2
Any other choices for θ1, θ2 make the likelihood of the
observed data slightly lower. But it’s actually pretty
unlikely that two sample points would be chosen
ˆ2
exactly equidistant from, and on opposite sides of the
mean, so the MLE
systematically underestimates θ2 .

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Maximum a posteriori
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Incorporating priors. How?
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MLE vs MAP estimation
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MLE for general problems
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Learning problem setting
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A set of random variables X from unknown distribution P*
Training data D = M instances of X: {d[1],…,d[M]}
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A parametric model P(X; Θ) (a ‘legal’ distribution)
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Define the likelihood function:
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L(Θ : D) =
Maximum likelihood estimation
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Choose parameters Θ* that satisfy:
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MLE for Bayesian networks
Structure G
x1
x2
PG = P(x1,x2,x3,x4)
= P(x1) P(x2) P(x3|x1,x2) P(x4|x1,x3)
More generally?
x3
x4
PG   P( x i | pai )
i
Parameters θ
Θx1, Θx2 , Θx3|x1,x2 , Θx4|x1,x3
(more generally Θxi|pai)
Given D: x[1],…x[m]…,x[M], estimate θ.
(x1[m],x2[m],x3[m],x4[m])

Likelihood decomposition:
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The local likelihood function for Xi is:
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Bayesian network with table CPDs
The Student example
The Thumbtack example
X
Intelligence Difficulty
vs
Grade
P(X)
Joint distribution
Parameters
Data
θ
D: {H…x[m]…T}
Likelihood function
L(θ:D) = P(D;θ)
MLE solution
P(I,D,G) =
θI, θD, θG|I,D
D: {(i1,d1,g1)…(i[m],d[m],g[m])…}
θMh(1-θ)Mt
θˆ 
Mh
Mh  Mt
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Maximum Likelihood Estimation Review
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Find parameter estimates which make observed
data most likely
General approach, as long as tractable likelihood
function exists
Can use all available information
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Example – Gene Expression

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Instruction for making the proteins
Instruction for when and where to make them
“Coding” Regions
“Regulatory” Regions (Regulons)

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Regulatory regions contain “binding sites” (6-20 bp).
“Binding sites” attract a special class of proteins, known as
“transcription
factors”.
What turns genes
on (producing a protein) and off?
Bound
factors
can initiate transcription (making RNA).
When istranscription
a gene turned
on or off?
Proteins
inhibit
canon?
also be bound to their
Where (inthat
which
cells)transcription
is a gene turned
binding
sites.
How many
copies of the gene product are produced?
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Regulation of Genes
Transcription Factor
(Protein)
RNA polymerase
(Protein)
DNA
CG..A
AC..T
Regulatory Element
(binding sites)
source: M. Tompa, U. of Washington
Gene
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Regulation of Genes
Transcription Factor
(Protein)
RNA polymerase
(Protein)
DNA
CG..A
AC..T
Regulatory Element
source: M. Tompa, U. of Washington
Gene
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Regulation of Genes
Transcription Factor
(Protein)
RNA
polymerase
DNA
CG..A
AC..T
Regulatory Element
source: M. Tompa, U. of Washington
Gene
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Regulation of Genes
Transcription Factor
DNA
CG..A
RNA
polymerase
AC..T
Regulatory Element
source: M. Tompa, U. of Washington
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New protein
The Gene regulation example
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What determines the expression level of a gene?
What are observed and hidden variables?
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e.G, e.TF’s: observed; Process.G: hidden variables  want to infer!
Expression level
of TF1
Biological process the
gene is involved in
e.TF1
e.TF2
e.TF3
e.TF4
...
e.TFN
Process.G
= p321
e.G
Expression level
of a gene
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The Gene regulation example


What determines the expression level of a gene?
What are observed and hidden variables?

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e.G, e.TF’s: observed; Process.G: hidden variables  want to infer!
How about BS.G’s? How deterministic is the sequence of a binding site?
How much do we know?
e.TF1
e.TF2
e.TF3
e.TF4
...
e.TFN
Process.G
BS1.G
...
= Yes
Whether the gene has
TF1’s binding site
BSN.G
= Yes
e.G
Expression level
of a gene
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Not all data are perfect

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Most MLE problems are simple to solve with
complete data.
Available data are “incomplete” in some way.
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Outline
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Learning from data
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Maximum likelihood estimation (MLE)
Maximum a posteriori (MAP)
Expectation-maximization (EM) algorithm
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Continuous space revisited
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Assuming sample x1, x2,…, xn is from a mixture of
parametric distributions,
x1 x2 … xm
xm+1 … xn
x
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A real example
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CpG content of human gene promoters
GC frequency
“A genome-wide analysis of CpG dinucleotides in the human genome distinguishes two 26
distinct classes of promoters” Saxonov, Berg, and Brutlag, PNAS 2006;103:1412-1417
Mixture of Gaussians
Parameters θ
means
variances
mixing parameters
P.D.F
L( 1 ,  2 ,  ,  , 1 , 2 : x1 ,..., xn )
2
1
2
2
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Apply MLE?
L( 1 ,  2 ,  12 ,  22 , 1 , 2 : x1 ,..., xn )
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No closed form solution known for finding θ
maximizing L.
However, what if we knew the hidden data?
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EM as Chicken vs Egg
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IF zij known, could estimate parameters θ
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e.g., only points in cluster 2 influence μ2, σ2.
IF parameters θ known, could estimate zij
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e.g., if |xi - μ1|/σ1 << |xi – μ2|/σ2, then zi1 >> zi2
Convergence provable? YES
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BUT we know neither; (optimistically) iterate:
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E-step: calculate expected zij, given parameters
M-step: do “MLE” for parameters (μ,σ), given E(zij)
Overall, a clever “hill-climbing” strategy
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“Classification EM”
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If zij < 0.5, pretend it’s 0; zij > 0.5, pretend it’s 1
i.e., classify points as component 0 or 1
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Now recalculate θ, assuming that partition
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Then recalculate zij , assuming that θ
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Then recalculate θ, assuming new zij , etc., etc.
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Full EM
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xi’s are known; Θ unknown. Goal is to find the MLE Θ of:
L (Θ : x1,…,xn )
(hidden data likelihood)
Would be easy if zij’s were known, i.e., consider
L (Θ : x1,…,xn, z11,z12,…,zn2 )
(complete data likelihood)
But zij’s are not known.
Instead, maximize expected likelihood of observed data
E[ L(Θ : x1,…,xn, z11,z12,…,zn2 ) ]
where expectation is over distribution of hidden data (zij’s).
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The E-step
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Find E(zij), i.e., P(zij=1)
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Assume θ known & fixed. Let
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A: the event that xi was drawn from f1
B: the event that xi was drawn from f2
D: the observed data xi
Then, expected value of zi1 is P(A|D)
P(A|D) =
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Complete data likelihood
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Recall:
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so, correspondingly,
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Formulas with “if’s” are messy; can we blend
more smoothly?
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M-step
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Find θ maximizing E[ log(Likelihood) ]
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EM summary
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Fundamentally an MLE problem
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Useful if analysis is more tractable when 0/1
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Hidden data z known
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Iterate:
E-step: estimate E(z) for each z, given θ
M-step: estimate θ maximizing E(log likelihood) given E(z)
where “E(logL)” is wrt random z ~ E(z) = p(z=1)
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EM Issues
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EM is guaranteed to increase likelihood with
every E-M iteration, hence will converge.
But may converge to local, not global, max.
Issue is intrinsic (probably), since EM is often
applied to NP-hard problems (including clustering,
above, and motif-discovery, soon)
Nevertheless, widely used, often effective
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Acknowledgement

Profs Daphne Koller & Nir Friedman,
“Probabilistic Graphical Models”

Prof Larry Ruzo, CSE 527, Autumn 2009

Prof Andrew Ng, ML lecture note
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