Maximum Likelihood Estimates and the EM Algorithms II

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# Maximum Likelihood Estimates and the EM Algorithms II - PowerPoint PPT Presentation

Maximum Likelihood Estimates and the EM Algorithms II. Henry Horng-Shing Lu Institute of Statistics National Chiao Tung University hslu@stat.nctu.edu.tw http://tigpbp.iis.sinica.edu.tw/courses.htm. Part 1 Computation Tools. Include Functions in R. source( “ file path ” ) Example

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### Maximum Likelihood Estimates and the EM Algorithms II

Henry Horng-Shing Lu

Institute of Statistics

National Chiao Tung University

hslu@stat.nctu.edu.tw

http://tigpbp.iis.sinica.edu.tw/courses.htm

### Part 1Computation Tools

Include Functions in R
• source(“file path”)
• Example
• In MME.R:
• In R:

### Part 2Motivation Examples

Example 1 in Genetics (1)

Two linked loci with alleles A and a, and B and b

A, B: dominant

a, b: recessive

A double heterozygote AaBb will produce gametes of four types: AB, Ab, aB, ab

A

a

b

B

A

b

a

a

A

A

a

b

B

B

B

b

F ( Female) 1- r’ r’ (female recombination fraction)

M (Male) 1-r r (male recombination fraction)

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Example 1 in Genetics (2)

r and r’ are the recombination rates for male and female

Suppose the parental origin of these heterozygote is from the mating of . The problem is to estimate r and r’ from the offspring of selfed heterozygotes.

Fisher, R. A. and Balmukand, B. (1928). The estimation of linkage from the offspring of selfed heterozygotes. Journal of Genetics, 20, 79–92.

http://en.wikipedia.org/wiki/Geneticshttp://www2.isye.gatech.edu/~brani/isyebayes/bank/handout12.pdf

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Example 1 in Genetics (4)

Four distinct phenotypes: A*B*, A*b*, a*B* and a*b*.

A*: the dominant phenotype from (Aa, AA, aA).

a*: the recessive phenotype from aa.

B*: the dominant phenotype from (Bb, BB, bB).

b* : the recessive phenotype from bb.

A*B*: 9 gametic combinations.

A*b*: 3 gametic combinations.

a*B*: 3 gametic combinations.

a*b*: 1 gametic combination.

Total: 16 combinations.

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Example 1 in Genetics (6)

Hence, the random sample of n from the offspring of selfed heterozygotes will follow a multinomial distribution:

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Example 1 in Genetics (7)

Suppose that we observe the data of

y = (y1, y2, y3, y4) = (125, 18, 20, 24),

which is a random sample from

Then the probability mass function is

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Maximum Likelihood Estimate (MLE)

Likelihood:

Maximize likelihood: Solve the score equations, which are setting the first derivates of likelihood to be zeros.

Under regular conditions, the MLE is consistent, asymptotic efficient and normal!

More:

http://en.wikipedia.org/wiki/Maximum_likelihood

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MLE for Example 1 (1)

Likelihood

MLE:

A B C

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MLE for Example 1 (2)

Checking:

(1)

(2)

(3)

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### Part 3Numerical Solutions for the Score Equations of MLEs

A Banach Space
• A Banach space B is a vector space over the field K such that
• Every Cauchy sequence of B converges in B (i.e., B is complete).

(http://en.wikipedia.org/wiki/Banach_space)

Lipschitz Continuous
• A closed subset and mapping
• F is Lipschitz continuous on A with if .
• F is a contraction mapping on A if F is Lipschitz continuous and

(http://en.wikipedia.org/wiki/Lipschitz_continuous)

Fixed Point Theorem
• If F is a contraction mapping on A if F is Lipschitz continuous and
• F has an unique fixed point such that
• initial , k=1,2,…

(http://en.wikipedia.org/wiki/Fixed-point_theorem)

(http://www.math-linux.com/spip.php?article60)

Parallel Chord Method (1)
• Parallel chord method is also called simple iteration.
Define Functions for Example 1 in R

We will define some functions and variables for finding the MLE in Example 1 by R

Newton-Raphson Method (1)
• http://math.fullerton.edu/mathews/n2003/Newton'sMethodMod.html
• http://en.wikipedia.org/wiki/Newton%27s_method
Halley’s Method
• The Newton-Raphson iteration function is
• It is possible to speed up convergence byusing more expansion terms than the Newton-Raphson method does when the object function is very smooth, like the method by Edmond Halley (1656-1742):

(http://math.fullerton.edu/mathews/n2003/Halley'sMethodMod.html)

Bisection Method (1)
• Assume that and that there exists a number such that . If and have opposite signs, and represents the sequence of midpoints generated by the bisection process, thenand the sequence converges to r.
• That is, .

(http://en.wikipedia.org/wiki/Bisection_method )

Secant Method

(http://en.wikipedia.org/wiki/Secant_method )

(http://math.fullerton.edu/mathews/n2003/SecantMethodMod.html )

S

C

B

A

Secant-Bracket Method
• The secant-bracket method is also called the regular falsi method.
Fisher Scoring Method
• Fisher scoring method replaces by where is the Fisher information matrix when the parameter may be multivariate.
Order of Convergence
• Order of convergence is p if

and c<1 for p=1.

(http://en.wikipedia.org/wiki/Order_of_convergence)

Note:

Hence, we can use regression to estimate p.

Theorem for Newton-Raphson Method
• If , F is a contraction mapping then p=1 and
• If exists, has a simple zero, then such that of the Newton-Raphson method is a contraction mapping and p=2.
Find Convergence Order by R (1)

R=Newton(y1, y2, y3, y4, initial)

#Newton method can be substitute for different method

temp=log(abs(R\$iteration-R\$phi));

y=temp[2:(length(temp)-1)]

x=temp[1:(length(temp)-2)]

lm(y~x)

Exercises

Write your own programs for those examples presented in this talk.

Write programs for those examples mentioned at the following web page:

http://en.wikipedia.org/wiki/Maximum_likelihood

Write programs for the other examples that you know.

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More Exercises (1)

Example 3 in genetics: The observed data are (nO, nA, nB, nAB) = (176, 182, 60, 17) ~ Multinomial(r^2, p^2+2pr, q^2+2qr, 2pq), where p, q, and r fall in [0,1] such that p+q+r = 1. Find the MLEs for p, q, and r.

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More Exercises (2)

Example 4 in the positron emission tomography (PET):The observed data are n*(d) ~Poisson(λ*(d)), d = 1, 2, …, D, and

The values of p(b,d) are known and the unknown parameters are λ(b), b = 1, 2, …, B.

Find the MLEs for λ(b), b = 1, 2, …, B.

.

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More Exercises (3)

Example 5 in the normal mixture:The observed data xi, i = 1, 2, …, n, are random samples from the following probability density function:

Find the MLEs for the following parameters:

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