Transcript Basic principles of probability theory

Maximum likelihood (ML)
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Conditional distribution and likelihood
Maximum likelihood estimator
Information in the data and likelihood
Observed and Fisher’s information
Home work
Introduction
It is often the case that we are interested in finding values of some parameters of the
system. Then we design an experiment and get some observations (x1,,,xn). We
want to use these observations and estimate the parameters of the system. Once
we know (it might be a challenging mathematical problem) how parameters and
observations are related then we can use this to estimate the parameters.
Maximum likelihood is one the techniques to estimate parameters using observations
or experimental data. There are other estimation techniques also. These include
Bayesian, least-squares, method of moments, M-estimators.
The result of the estimation is a function of observation – t(x1,,,xn). A function of the
observations is called statistic. It is a random variable and in many cases we
want to find its distribution also. In general, finding the distribution of the
statistic is a challenging problem. But there are numerical technique (e.g.
bootstrap) to approximate this distribution.
Desirable properties of an estimation
1)
Unbiasedness. Bias is defined as a difference between estimator (t) and true
parameter (). Expectation is taken using the probability distribution of
observations
bias  E (t)  
2)
3)
4)
Efficiency. Efficient estimation is that with minimum variance (var(t)=E(tE(t))2). Efficiency of the estimator is measured by its variance.

Consistency.
If the number of observations goes to infinity then an estimator
converges to true value then this estimator is called a consistent estimator
| tn  | 0
Minimum mean square error (Minimum m.s.e). M.s.e. is defined as the
expectation value of the square of the difference (error) between estimator and
the true value
m.s.e.  E(t   ) 2  var( t)  (E(t)   ) 2  var( t)  bias 2
Minimum m.s.e. means that this estimator must be efficient and unbiased. It is very
difficult toachieve all these properties. Under some conditions ML estimator
obeys them asymptotically. Moreover the distribution of ML estimator is
asymptotically normal that simplifies the interpretation of the results.
Conditional probability distribution and likelihood
Let us assume that we know that our random sample points came from a population
with the distribution with parameter(s) - . We do not know . If we would
know it, then we could write the probability distribution of a single observation
f(x|). Here f(x|) is the conditional distribution of the observed random
variable if the parameter(s) would be known. If we observe n independent
sample points from the same population then the joint conditional probability
distribution of all observations can be written:
n
f ( x1 , x2 , , , xn |  )   f ( xi |  )
i 1
We could write the product of the individual probability
distributions because the
observations are independent (independent conditionally when parameters are
known). f(x|) is the probability mass function of an observation for discrete
and density of the distribution for continuous cases.
We could interpret f(x1,x2,,,xn|) as the probability of observing given sample points if
we would know the parameter . If we would vary the parameter(s) we would
get different values for the probability f. Since f is the probability distribution,
parameters are fixed and observation varies. For a given set of observations we
define likelihood proportional to the conditional probability distribution.
L( x1 , x2 , , , xn |  )  f ( x1 , x2 , , , xn |  )
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 observations vary.
When we talk about likelihood then observations are fixed parameters vary.
That is the major difference between likelihood and conditional probability
distribution. Sometimes to emphasize that parameters vary and observations are
fixed, likelihood is written as:
L( | x1 , x2 , , , , xn )
In this and following lectures we will use one notation for probability and likelihood.
When we talk about probability then we assume that observations vary and
when we talk about likelihood we assume that parameters vary.
Principle of maximum likelihood states that the best parameters are those that
maximise probability of observing current values of the observations.
Maximum likelihood chooses parameters that satisfy:
L( x1, x2 , , , xn | ˆ)  L( x1, x2 , , , xn |  )
Maximum likelihood
Purpose of the maximum likelihood is to maximize the likelihood function and
estimate parameters. If the derivatives of the likelihood function exist then it
can be done using:
dL ( x1 , x2 , , , xn |  )
0
d
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 maximized. Since log is strictly
monotonically increasing function, derivative of the likelihood and derivative
of the log of likelihood will have exactly same roots. If we use the fact that
observations are independent then the joint probability distributions of all
observations is equal to the product of the individual probabilities. We can
write log of the likelihood (denoted as l):
n
l ( x1 , x2 , , , , xn |  )  ln(L( x1 , x2 , , , , xn |  )   ln( f ( xi |  ))
i 1
Usually working with sums is easier than working with products
Likelihood: Normal distribution example
Let us assume that our observations come from the
population with N(0,1). We have five observations.
For each obervation we can write loglikelihood
function (red lines). Loglikelihood function for all
observations is the sum of individual loglikelihood
functions (black line). As it can be seen likelihood
function for five observations combined has much
more pronounced maximum than that for individual
observations. Usually more observations we have
from the same population better is the estimation of
the parameter.
Likelihood: Binomial distribution example
Now let us take 10 observations from binomial distributions with size=1 (i.e. we do only
one trial). Let us assume that probability of success is equal to 0.5. Since each observation
is either 0 or 1 loglikelihood function for individual observation will be one of the two
functions (red lines on the left figure). Product of individual loglikelihood functions has
well defined maximum. Although logglikelihood function has flat maximum, the
likelihood function (right figure) has very well pronounced maximum.
Loglikelihood function
Likelihood function for five observations,
normalised to make the integral equal to one
Maximum likelihood: Example – success and failure
Let us consider two examples of estimation using maximum likelihood. 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 the value of . Let us
assume we have n trials and k of them are successes and n-k of them are failures.
Values of random variables in our trials can be 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:
f ( yi |  )   yi (1   )1 yi
Since individual trials are independent we can
write for n trials:
n
L( y1 , y2 , , , yn |  )    yi (1   )1 yi
i 1
log of this function is:
n
l ( y1 , y2 , , , yn |  )   ( yi ln( )  (1  yi ) ln(1   ))
i 1
Equating the first derivative
of the
likelihood w.r.tn unknown parameter to zero we get:
n
n
dl 
 i 1
d

yi

 (1  y )
i 1
i
1
 0  ˆ 
y
i 1
n
i

k
n
The ML estimator for the parameter is equal to the fraction of successes.
Maximum likelihood: Example – success and failure
In the example of successes and failures the result was not unexpected and we could
have guessed it intuitively. More interesting problems arise when parameter 
itself becomes function of some other parameters. Let us say:
  (  x)
The most popular form of the function
1  is logistic curves:

(z) 
z
1 e
, z    x
If for each trial x takes different value nthen the log likelihood function looks like:
l ( y1, y2 , , , yn , x1, x2 , , , xn |  ,  )   yi ln (  xi )  (1  yi ) ln(1  (  xi ))
i 1
Finding maximum of this function is more complicated. This problem can be

considered as a non-linear optimization problem. This kind of problems are
usually solved iteratively. I.e. a solution to the problem is guessed and then it is
improved iteratively. We will come back to this problem in the lecture on
generalised linear models
Logistic curve
Maximum likelihood: Example – normal distribution
Now let us assume that the sample points came from the population with normal
distribution with unknown mean and variance. Let us assume that we have n
observations, y=(y1,y2,,,yn). We want to estimate the population mean and
variance. Then log likelihood function will have the form:
n
n
1
( yi   )2
n
( yi   )2
2
2
l ( y1, y2 , , , yn | ,  )   ln(
e(
))  n ln( 2 )  ln( )  
2
2
2

2
2 2
2
i 1
i 1
If we get derivative of this function w.r.t mean
value and variance then we can write:
n
dl
1
 2
d 
n
( y
i 1
i
  )  0  ˆ 
y
i 1
i
n
 ˆ  y
dl
n
n n
 2 
( yi   )2  0
2
4 
d ( )
2
2 i 1
Fortunately first of these equations can be solved without knowledge about the
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:
1 n
ˆ  (y i  
ˆ )2

n i1
2
Maximum likelihood: Example – normal distribution
Maximum likelihood estimator in this case gave a 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 the mean value. In other words  is a function of
parameters of interest.
  g(x1, x 2 ,,,x n ,1, 2,,, k )
Then the problem is to estimate parameters using maximum likelihood estimator.
Usually x-s are fixed values (fixed effects model). When x-s are random
(random or mixed effect models) then the treatment becomes more

complicated.
We will have one lecture on mixed effect models.
Parameters are -s. If this function is linear on parameters then we have linear
regression.
If variances are known then the Maximum likelihood estimator using observations
with normal distribution becomes least-squares estimator.
Maximum likelihood: Example – normal distribution
If all σ-s are equal to each other and our interest is only in estimation of mean value
(μ) then minus loglikelihood function, after multiplying by σ2 and igonring all
constants that do not depend on mean value, can be written:
1 n
LSQ(y1, y 2 ,,,y n | )  (y i  ) 2
2 i1
It is the most popular estimator - least-squares function. If we consider the central
limit theorem then we can say that in many cases distributions of the errors in

the observations
can be approximated with normal distribution and that explain
why this function is so popular. It is a special case of maximum likelihood
estimators.
We will come back to this function in linear model lecture.
Information matrix: Observed and Fisher’s
One of the important aspects of a likelihood function is its behavior near to the 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 the likelihood has a pronounced peak then
small changes of the parameters would cause large changes in the probability. In this
cases we say that observation has more information about parameters. It is usually
expressed as the second derivative (or curvature) of the minus log-likelihood function.
Observed information is equal to the second derivative of the minus log-likelihood
function:
d 2l ( y |  )
I o ( )  
d 2
When there are more than one parameter it is called information matrix.
Usually it is calculated at the maximum of the likelihood.
Example: In case of successes and failures we can write:
n
I o ( ) 
 yi
i 1
2
n

 (1  y )
i 1
i
(1   )2
N.B. Note that it is one of the definitions of information.
Information matrix: Observed and Fisher’s
Expected value of the observed information matrix is called expected information
matrix or Fisher’s information. Expectation is taken over observations:
I ( )  E ( I o ( ))
It is calculated at any value of the parameter. Interesting fact about Fisher’s
information matrix is that it is also equal to the expected value of the product of
the gradients of loglikelihood function:
d 2l ( y |  )
dl( y |  ) dl( y |  )
I ( )   E (
)

E
(
)
d 2
d
d
Note that observed information depends on particular values of the observations
whereas expected information depends only on the probability distribution of
the observations (It is a result of integration. When we integrate over some
variables we loose dependence on particular values):
When sample size becomes large then maximum likelihood estimator becomes
approximately normally distributed with the variance close to :
I 1( ) or Io1( )
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 Fisher’s
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 expected information:
var( )  I 1( )
Information matrix: Observed and Fisher’s
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:
n
n
n
n
yi  (1  yi )

d 2l ( y |  )
I ( )  E ( 
)  E ( i 1 2  i 1
)
2
d

(1   )2
 E ( y )  E (1  y )
i 1
i

2

i
i 1
(1   )
2

n

2

n(1   )
n

2
(1   )
 (1   )
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:
var( ˆ ) 
ˆ (1  ˆ )
n
This statement is true for large sample sizes.
Information matrix and distribution of parameters: Example
Distribution of parameter of the interest can be derived using Bayes’s theorem and
assuming that we have no information about the parameter before the
observations are made.
f (;x1,...,x n )  f ( ) f (x1,...,x n ; ) / f (x1,...,x n )
If we assume that f(β) is constant then the distribution of parameter can be derived by

renormalisation
of the conditional probability distribution of observations given
parameter(s) is known.
f (;x1,...,xn ) 
1
f (x1,...,x n ; )
N
Let us compare for binomial distributions (with the number of trials 1, the number of
observations 50 and probability of success 0.5) normal approximation and the

distribution
itself. Mean value is 0.46, standard deviation of normal
approximation derived using information matrix is 0.0705.
Black line actual distribution and red line normal
approximation. For this case asymptotic
distribution almost exactly coincides with the actual
distribution
References
1.
2.
Berthold, M. and Hand, DJ (2003) “Intelligent data analysis”
Stuart, A., Ord, JK, and Arnold, S. (1991) Kendall’s advanced
Theory of statistics. Volume 2A. Classical Inference and the Linear
models. Arnold publisher, London, Sydney, Auckland
Exercise 1
a) Assume that we have a sample of size n (x1, x2, …,xn) independently drawn from
the population with the density of probability distribution (it is exponential
distribution where  has been replaced by 1/  )
f (x | ) 
1

ex /  0  x  
Find the maximum likelihood estimator for . Show that maximum likelihood
estimator is unbiased. Find the observed and expected information?

b) Negative binomial distribution has the form:
f (k | p)  pk (1 p), where k= 0,1,2....
This is the probability mass function for the number of successful trials before a
failure occurs. Probability of success is p and that of failure is 1-p. If we have a
sample of n independent points – (k1,k2,….,kn) drawn from the population with

the negative binomial distribution (in i-th time we had ki successes and (ki+1)-st
trial was failure). Find the maximum likelihood estimator for p. What is the
observed and expected information?