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Maximum Likelihood ( ML ) Parameter Estimation with applications to reconstructing phylogenetic trees Comput. Genomics, lecture 6b. Presentation taken from Nir Friedman’s HU course, available at www.cs.huji.ac.il/~pmai . Changes made by Dan Geiger, Ydo Wexler, and finally by Benny Chor. - PowerPoint PPT Presentation
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.
Maximum Likelihood (ML) Parameter Estimation
with applications to reconstructing phylogenetic
trees
Comput. Genomics, lecture 6b
Presentation taken from Nir Friedman’s HU course, available at www.cs.huji.ac.il/~pmai.
Changes made by Dan Geiger, Ydo Wexler, and finally by Benny Chor.
2
The Setting We have a probabilistic model, M, of some
phenomena. We know exactly the structure of M, but not the values of its probabilistic parameters, .
Each “execution” of M produces an observation, x[i] , according to the (unknown) distribution induced by M.
Goal: After observing x[1] ,…, x[n] , estimate the model parameters, , that generated the observed data.
3
Maximum Likelihood Estimation (MLE)
The likelihood of the observed data, given the model parameters , as the conditional probability that the model, M, with parameters , produces x[1] ,…, x[n] .
L()=Pr(x[1] ,…, x[n] | , M),
In MLE we seek the model parameters, , that maximize the likelihood.
4
Maximum Likelihood Estimation (MLE) In MLE we seek the model parameters, , that maximize the likelihood. The MLE principle is applicable in a wide variety of applications, from speech recognition, through natural language processing, to computational biology.
We will start with the simplest example: Estimating the bias of a coin. Then apply MLE to inferring phylogenetic trees. (will later talk about MAP - Bayesian inference).
5
Example: Binomial Experiment
When tossed, it can land in one of two positions: Head (H) or Tail (T)
Head Tail
We denote by the (unknown) probability P(H).
Estimation task: Given a sequence of toss samples x[1], x[2], …, x[M] we want to estimate the probabilities P(H)= and P(T) = 1 -
6
Statistical Parameter Fitting (restement)
Consider instances x[1], x[2], …, x[M]
such that The set of values that x can take is known Each is sampled from the same distribution Each sampled independently of the rest
i.i.d.Samples(why??)
The task is to find a vector of parameters that have generated the given data. This vector parameter can be used to predict future data.
7
The Likelihood Function How good is a particular ?
It depends on how likely it is to generate the observed data
The likelihood for the sequence H,T, T, H, H is
( ) ( | ) ( [ ] | )Dm
L P D P x m
( ) (1 ) (1 )DL
0 0.2 0.4 0.6 0.8 1
L()
8
Sufficient Statistics
To compute the likelihood in the thumbtack example we only require NH and NT
(the number of heads and the number of tails)
NH and NT are sufficient statistics for the binomial distribution
( ) (1 )HD
TN NL
9
Sufficient Statistics
A sufficient statistic is a function of the data that summarizes the relevant information for the likelihood
Datasets
Statistics
Formally, s(D) is a sufficient statistics if for any two datasets D and D’
s(D) = s(D’ ) LD() = LD’ ()
10
Maximum Likelihood Estimation
MLE Principle:
Choose parameters that maximize the likelihood function
This is one of the most commonly used estimators in statistics
Intuitively appealing One usually maximizes the log-likelihood
function, defined as lD() = ln LD()
11
Example: MLE in Binomial Data
Taking derivative and equating it to 0,
we get
log log 1D H Tl N N
1H TN N
0 0.2 0.4 0.6 0.8 1
L()
Example:(NH,NT ) = (3,2)
MLE estimate is 3/5 = 0.6
ˆ H
H T
N
N N
(which coincides with what one would expect)
12
From Binomial to Multinomial
Now suppose X can have the values 1,2,…,K (For example a die has K=6 sides)
We want to learn the parameters 1, 2. …, K
Sufficient statistics:N1, N2, …, NK - the number of times each outcome is observed
Likelihood function:
MLE: (proof @ assignment 3)
1
( ) k
KN
D kk
L
ˆ kk
N
N
13
Example: Multinomial
Let be a protein sequence We want to learn the parameters q1, q2,…,q20
corresponding to the frequencies of the 20 amino acids
N1, N2, …, N20 - the number of times each amino acid is observed in the sequence
Likelihood function:20
1
( ) kND k
k
L q q
1 2.... nx x x
kk
Nq
nMLE:
14
Inferring Phylogenetic Trees
Let be n sequence (DNA or AA).
Assume for simplicity they are all same length, l. We want to learn the parameters of a
phylogenetic tree that maximizes the likelihood.
But wait: Should first specify a model.
1 2, ,.... , nS S S
15
A Probabilistic Model
Our models will consist of a “regular” tree, where
in addition, edges are assigned substituion probabilities.
For simplicity, assume our “DNA” has only two
states, say X and Y. If edge e is assigned probability pe , this means
that the probability of substitution (X Y)
across e is pe .
16
A Probabilistic Model (2)
Our models will consist of a “regular” tree, where
in addition, edges are assigned substituion probabilities.
For simplicity, assume our “DNA” has only two
states, say X and Y. If edge e is assigned probability pe , this means
that the probability of substitution (X Y)
across e is pe .
17
A Probabilistic Model (3) If edge e is assigned probability pe , this means
that the probability of more involved patterns of
substitution across e (e.g. XXYXY YXYXX)
is determined, and easily computed: pe2
(1- pe)3
for this pattern. Q.: What if pattern on both sides is known, but pe is
not known? A.: Makes sense to seek pe that maximizes
probability of observation. So far, this is identical to coin toss example.
18
A Probabilistic Model (4)
Now we don’t know the states at internal node(s), nor
the edge parameters pe1, pe2, pe3
XXYXY YXYXX
YYYYX
pe1
pe2
pe3
But a single edge is a fairly boring tree…
?????
19
Two Ways to Go
1. Maximize over states of internal node(s)
2. Average over states of internal node(s)
In both cases, we maximize over edge parameters
XXYXY YXYXX
YYYYX
pe1
pe2
pe3?????
20
Two Ways to Go
In the first version (average, or sum over states of internal
nodes) we are looking for the “most likely” setting of tree edges.
This is called maximum likelihood (ML) inference of
phylogenetic trees.
ML is probably the inference method most widely
(some would say wildly ) used.
XXYXY YXYXX
YYYYX
pe1
pe2
pe3?????
21
Two Ways to Go
In the second version (maximize over states of
internal nodes) we are looking for the “most likely”
ancestral states. This is called ancestral maximum
likelihood (AML).
In some sense AML is “between” MP (having
ancestral states) and ML (because the goal is still to
maximize likelihood).
XXYXY YXYXX
YYYYX
pe1
pe2
pe3?????
22
Back to the Probabilistic Model
A reconstruction method is called statistically consistent if the model it reconstructs converges to
the “true tree” as the length of the sequences goes
to infinity.
XXYXYYXYXY
YYYYX
pe1
pe2
pe3
pe4
XXXYX
pe5
23
Consistency, and Beyond A reconstruction method is called statistically consistent if the model it reconstructs converges to the “true tree” as the length of the sequences goes to infinity.
We would like a reconstruction method that is
(1)Statistically Consistent(2)Computationally efficient