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Maximum likelihood

- Conditional distribution and likelihood
- Maximum likelihood estimations
- Information in the data and likelihood
- Observed and Fishers information
- Exercise

Conditional probability distribution and

likelihood

- Let us assume that we know that our random sample

points come from the distribution with

parameter(s) ?. We do not know ?. If we would

know it then we could write probability

distribution of single observation f(x?). Here

f(x?) is conditional distribution of the

observed random variable if parameter would be

known. If we observe n independent sample points

from the same population then joint conditional

probability distribution of all observations can

be written - We could write product of the individual

probability distribution because observations are

independent. f(x?) is probability of observation

for discrete cases and density of the

distribution for continuous cases. - We could interpret f(x1,x2,,,xn?) as probability

of observing given sample points if we would know

parameter ?. If would vary the parameter we would

get different values for the probability f. Since

f is the probability distribution, parameters are

fixed and observation varies. For a given

observation we define likelihood equal to the

conditional probability distribution.

Conditional probability distribution and

likelihood Cont.

- When we talk about conditional probability

distribution of the observations given

parameter(s) then we assume that parameters are

fixed and observation vary. When we talk about

likelihood then observations are fixed parameters

vary. That is major difference between likelihood

and conditional probability distribution.

Sometimes to emphasize that parameters vary and

observations are fixed, likelihood is written as - In this and following lectures we will use one

notation for probability and likelihood. When we

will talk about probability then we will assume

that observations vary and when we will talk

about likelihood we will assume that parameters

vary. - Principle of maximum likelihood states that best

parameters are those that maximise probability of

observing current values of observations. Maximum

likelihood chooses parameters that satisfy

Maximum likelihood

- Purpose of maximum likelihood is to maximise

likelihood function and estimate parameters. If

derivatives of the likelihood function exist then

it can be done using - Solution of this equation will give possible

values for maximum likelihood estimator. If the

solution is unique then it will be the only

estimator. In real application there might be

many solutions. - Usually instead of likelihood its logarithm is

maximised. Since log is monotonically increasing

function, derivative of likelihood and derivative

of the log of likelihood will have exactly same

roots. If we use the fact that observations are

independent then joint probability distribution

of all observations is equal to product of

individual probabilities. We can write log of

likelihood (denoted as l) - Usually working with sums is easier than working

with products

Maximum likelihood Example success and failure

- Let us consider two examples. First example

corresponds to discrete probability distribution.

Let us assume that we carry out trials. Possible

outcomes of the trials are success or failure.

Probability of success is ? and probability of

failure is 1- ?. We do not know value of ?. Let

us assume we have n trials and k of them are

successes and n-k of them are failures. Value of

random variable describing our trials are either

0 (failure) or 1 (success). Let us denote

observations as y(y1,y2,,,,yn). Probability of

the observation yi at the ith trial is - Since individual trials are independent we can

write for n trials - For log of this function we can write
- Derivative of the likelihood wrt unknown

parameter is - Estimate for the parameter is equal to fraction

of successes.

Maximum likelihood Example success and failure

- .In the example of successes and failure result

was not unexpected and we could have guessed it

intuitively. More interesting problems arise when

parameter ? itself becomes function of some other

parameters and possible observations also. Let us

say - It may happen that xi themselves are random

variable also. If it is case and the function

corresponds to normal distribution then analysis

is called Probit analysis. Then log likelihood

function would look like - Finding maximum of this function is more

complicated. This problem can be considered as a

non-linear optimisation problem. This kind of

problems are usually solved iteratively. I.e.

solution to the problem is guessed and then it is

improved iteratively.

Maximum likelihood Example normal distribution

- Now let us assume sample points come from the

population with normal distribution with unknown

mean and variance. Let us assume that we have n

observation, y(y1,y2,,,yn). We want to estimate

population mean and variance. Then log likelihood

function will have the form - If we get derivative of this function w.r.t mean

value and variance then we can write - Fortunately first of these equations can be

solved without knowledge about second one. Then

if we use result from the first solution in the

second solution (substitute ? by its estimate)

then we can solve second equation also. Result of

this will be sample variance

Maximum likelihood Example normal distribution

- Maximum likelihood estimator in this case gave

sample mean and sample variance. Many statistical

techniques are based on maximum likelihood

estimation of the parameters when observations

are distributed normally. All parameters of

interest are usually inside mean value. In other

word ? is a function of several parameters. - Then problem is to estimate parameters using

maximum likelihood estimator. Usually either x-s

are fixed values or random variables. Parameters

are ?-s. If this function is linear then we have

linear regression.

Information matrix Observed and Fishers

- One of the important aspects of the likelihood

function is its behavior near to maximum. If the

likelihood function is flat then observations

have little to say about the parameters. It is

because changes of the parameters will not cause

large changes in the probability. That is to say

same observation can be observed with similar

probabilities for various values of the

parameters. On the other hand if likelihood has

pronounced peak near to the maximum then small

changes in parameters would cause large changes

in probability. In this cases we say that

observation has more information about

parameters. It is usually expressed as second

derivative of the log-likelihood function.

Observed information is equal second derivative

of the minus log-likelihood function - Usually it is calculated at the maximum of the

likelihood. This information is different from

that defined using entropy. - Example In case of successes and failures we can

write

Information matrix Observed and Fishers

- Expected value of the observed information matrix

is called expected information or Fishers

information matrix. Expectation is taken over

observations - It is calculated at any value of the parameter.

Remarkable fact about Fishers information matrix

is that it is also equal expected value of

product of the gradients (first derivatives) - Note that observed information matrix depends on

particular observation whereas expected

information matrix depends only on probability

distribution of observations (It is result of

integration. When integrate over variables we

loose dependence on these variables) - When sample size becomes large then maximum

likelihood estimate becomes approximately

normally distributed with variance close to - Fisher points out that inversion of observed

information matrix gives slightly better estimate

to variance than that of the expected information

matrix.

Information matrix Observed and Fishers

- More precise relation between expected

information and variance is given by Cramer and

Rao inequality. According to this inequality

variance of the maximum likelihood estimator

never can be less than inversion of information - Now let us consider an example of successes and

failures. If we get expectation value for the

second derivative of minus log likelihood

function we can get - If we take this at the point of maximum

likelihood then we can say that variance of the

maximum likelihood estimator can be approximated

by - This statement is true for large sample sizes.

Exercise 1

- a) Assume that we have n sample points

independently drawn from population with

exponential distribution - What is maximum likelihood estimator for ?.
- b) Now consider the case when population has the

distribution - What is maximum likelihood estimator for ?. What

is observed information for ?. What is expected

information for ?.

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