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#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\use_default_options true
\begin_modules
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\language english
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\end_header
\begin_body
\begin_layout Title
An Example of Maximum Likelihood Estimation
\end_layout
\begin_layout Author
Jeff Laake
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
<<echo=F>>=
\end_layout
\begin_layout Plain Layout
llb=function(n,N,p){n*log(p)+(N-n)*log(1-p)}
\end_layout
\begin_layout Plain Layout
ndllb=function(n,N,p,delta){(llb(n,N,p+delta)-llb(n,N,p-delta))/(2*delta)}
\end_layout
\begin_layout Plain Layout
nd2llb=function(n,N,p,delta){(llb(n,N,p+delta)+llb(n,N,p-delta)-2*llb(n,N,p))/(d
elta^2)}
\end_layout
\begin_layout Plain Layout
adllb=function(n,N,p){n/p-(N-n)/(1-p)}
\end_layout
\begin_layout Plain Layout
ad2llb=function(n,N,p){-n/p^2-(N-n)/(1-p)^2}
\end_layout
\begin_layout Plain Layout
logodds=function(p,sep)
\end_layout
\begin_layout Plain Layout
{
\end_layout
\begin_layout Plain Layout
C=exp(1.96*sep/(p*(1-p)))
\end_layout
\begin_layout Plain Layout
return(list(lower=p/(p+(1-p)*C),upper=p/(p+(1-p)/C)))
\end_layout
\begin_layout Plain Layout
}
\end_layout
\begin_layout Plain Layout
p0=0.82
\end_layout
\begin_layout Plain Layout
setp=0.45
\end_layout
\begin_layout Plain Layout
N=100
\end_layout
\begin_layout Plain Layout
epsilon=0.0001
\end_layout
\begin_layout Plain Layout
@
\end_layout
\end_inset
\end_layout
\begin_layout Standard
A likelihood function is any function of
\begin_inset Formula $\theta$
\end_inset
(one or more parameters) that is proportional to the probability density
function (pdf)
\begin_inset Formula $f(z|\theta)$
\end_inset
of the data
\begin_inset Formula $z$
\end_inset
.
Maximum likelihood estimation (MLE) is a method of estimation that produces
the point estimate for
\begin_inset Formula $\theta$
\end_inset
that maximizes the likelihood function which means that the estimate is
the most likely value given the data at hand.
Consider a simple example of flipping a coin
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N}
\end_layout
\end_inset
times in which the outcome is
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N*setp}
\end_layout
\end_inset
heads.
The pdf is a binomial distribution with
\begin_inset Formula $N$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N}
\end_layout
\end_inset
and n=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N*setp}
\end_layout
\end_inset
heads:
\begin_inset Formula
\[
f(z=45|\theta)=\frac{100!}{45!55!}\theta^{45}(1-\theta)^{55}
\]
\end_inset
\end_layout
\begin_layout Standard
\noindent
The more typical formula using
\begin_inset Formula $p$
\end_inset
in place of
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\theta$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\uuline default
\uwave default
\noun default
\color inherit
and
\begin_inset Formula $n$
\end_inset
in place of
\begin_inset Formula $z$
\end_inset
is:
\begin_inset Formula
\[
f(n|p)=\frac{N}{n!(N-n)!}\, p^{n}(1-p)^{N-n}
\]
\end_inset
\end_layout
\begin_layout Standard
\noindent
Techincally,
\begin_inset Formula $\theta$
\end_inset
could include both
\begin_inset Formula $N$
\end_inset
and
\begin_inset Formula $p$
\end_inset
but in this example
\begin_inset Formula $N$
\end_inset
is known and fixed and thus only
\begin_inset Formula $p$
\end_inset
must be estimated.
The binomial coefficient (first part of the pdf) is constant with respect
to
\begin_inset Formula $p$
\end_inset
, so we can drop this to write the following likelihood function:
\begin_inset Formula
\[
L(p|n)=\, p^{n}(1-p)^{N-n}
\]
\end_inset
which is now specified as a function of
\begin_inset Formula $p$
\end_inset
which is conditional on the value of
\begin_inset Formula $n$
\end_inset
(
\begin_inset Formula $N$
\end_inset
as well but it is a constant and not data).
If you were to compute the values of the likelihood function they would
be quite small numerically if
\begin_inset Formula $n$
\end_inset
and
\begin_inset Formula $N$
\end_inset
are very large.
For example, with
\begin_inset Formula $n$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N*setp}
\end_layout
\end_inset
and
\begin_inset Formula $N$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N}
\end_layout
\end_inset
, the values range from
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{formatC(min(exp(llb(setp*N,N,(1:99)/100))),format="e",digits=3)}
\end_layout
\end_inset
to
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{formatC(max(exp(llb(setp*N,N,(1:99)/100))),format="e",digits=3)}
\end_layout
\end_inset
for values of
\begin_inset Formula $p$
\end_inset
from 0.01 to 0.99.
Very small numbers can become problematic with the limitations of computer
numerical precision, so typically we use use the natural logarithm (base
e) of the likelihood function (log-likelihood).
The logarithm of a product is the sum of the logarithms.
An exponential
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $p^{n}$
\end_inset
is a shorthand for a product (product of
\begin_inset Formula $p$
\end_inset
for
\begin_inset Formula $n$
\end_inset
times) so
\begin_inset Formula $\log(p^{n})=n\:\log(p)$
\end_inset
(the sum of
\begin_inset Formula $\log(p)$
\end_inset
for
\begin_inset Formula $n$
\end_inset
times) and the general log-likelihood for a binomial distribution is:
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula
\[
log(L(p|n))=n\log(p)+(N-n)\log(1-p)
\]
\end_inset
which for the same range of values for
\begin_inset Formula $p$
\end_inset
ranges from
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{formatC(min(llb(setp*N,N,(1:99)/100)),format="f",digits=3)}
\end_layout
\end_inset
to
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{formatC(max(llb(setp*N,N,(1:99)/100)),format="f",digits=3)}
\end_layout
\end_inset
.
\end_layout
\begin_layout Standard
To help understand how MLEs are constructed, look at the following plot
which demonstrates the shape of the log-likelihood as a function of
\begin_inset Formula $p$
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
<<echo=F,fig=TRUE>>=
\end_layout
\begin_layout Plain Layout
p=(1:99)/100
\end_layout
\begin_layout Plain Layout
plot(p,llb(setp*N,N,p),type="b",ylab="Log-likelihood")
\end_layout
\begin_layout Plain Layout
@
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
The log-likelihood provides a measure to assess which value of p is the
most likely given the observed data (i.e.,
\begin_inset Formula $n$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp*N}
\end_layout
\end_inset
out of
\begin_inset Formula $N$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N}
\end_layout
\end_inset
flips).
As you can see, it appears to have a maximum value something around 0.5.
If we trim the set of values we can get a better idea at what value of
p the maximum is attained.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
<<echo=F,fig=TRUE>>=
\end_layout
\begin_layout Plain Layout
p=(setp*N-10):(setp*N+10)/100
\end_layout
\begin_layout Plain Layout
plot(p,llb(setp*N,N,p),type="b",ylab="Log-likelihood")
\end_layout
\begin_layout Plain Layout
@
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
It appears that the maximum is at
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp}
\end_layout
\end_inset
.
In fact,
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp}
\end_layout
\end_inset
is the maximum likelihood estimate as shown later.
\end_layout
\begin_layout Standard
The MLE is the value of the parameter at which the log-likelihood is at
the maximum value (peak) or at the minimum (valley) of the negative log-likelih
ood.
At maximum or minimum, the derivative (the rate of change) is 0 and as
you move from one side of the peak (or valley) the derivative changes sign
(positive to negative or vice-versa).
In this example, the derivative changes sign from positive to negative
passing through 0 at
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp}
\end_layout
\end_inset
as shown in the following plot:
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
<<echo=F,fig=TRUE>>=
\end_layout
\begin_layout Plain Layout
p=(setp*N-10):(setp*N+10)/100
\end_layout
\begin_layout Plain Layout
plot(p,N*setp/p-N*(1-setp)/(1-p),type="b",ylab="Derivative of Log-likelihood")
\end_layout
\begin_layout Plain Layout
abline(h=0)
\end_layout
\begin_layout Plain Layout
abline(v=setp)
\end_layout
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
@
\end_layout
\end_inset
\end_layout
\begin_layout Standard
The derivative for this example can be computed analytically.
Remember from calculus, the derivative of log(p) with respect to p is 1/p
and for log(1-p) it is -1/(1-p)):
\end_layout
\begin_layout Standard
\noindent
\begin_inset Formula
\[
\frac{\partial log(L(p|n))}{\partial p}=\frac{n}{p}-\frac{N-n}{1-p}
\]
\end_inset
which is 0 for
\begin_inset Formula $n$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp*N}
\end_layout
\end_inset
,
\begin_inset Formula $N$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N}
\end_layout
\end_inset
and
\begin_inset Formula $p$
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp}
\end_layout
\end_inset
.
The closed-form MLE is derived by setting the derivative to 0 and solving
for
\begin_inset Formula $p$
\end_inset
which yields the estimator
\begin_inset Formula $n/N$
\end_inset
.
Typically, the estimator is denoted with a ^ to distinguish it from the
true parameter, as in
\begin_inset Formula $\hat{p}=\nicefrac{n}{N}$
\end_inset
.
In this case the estimator yields the value
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp*N}
\end_layout
\end_inset
/
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{N}
\end_layout
\end_inset
=
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Sexpr{setp}
\end_layout
\end_inset
.
In most realistic cases, a closed-form maximum likelihood estimator cannot
be derived and the estimate must be obtained by numerical optimization
which algorithmically searches the surface to find the maximum (peak) or
minimum (valley) for the negative log-likelihood.
\end_layout
\begin_layout Standard
Newton's method is one approach to numerical optimization which was devised
to find the root of an equation which is the value at which a function
value is 0.
In this case, the function is the derivative of the log-likelihood and
the MLE is the root (i.e, the value at which the derivative is 0).
Numerical optimization methods are typically iterative in which you take
a guess at the value, compute the function and a direction, obtain a new
value and then repeat the process until you are sufficiently close to the
result.
If
\begin_inset Formula $f(x)$
\end_inset
is our function then let
\begin_inset Formula $f'(x)$
\end_inset
be the derivative with respect to
\begin_inset Formula $x$
\end_inset
.
In this case,
\begin_inset space ~
\end_inset
\begin_inset Formula $f(x)$
\end_inset
is the first derivative of the log-likelihood and
\begin_inset Formula $f'(x)$
\end_inset
is the second derivative of the log-likelihood.
Replacing
\begin_inset Formula $x$
\end_inset
with
\begin_inset Formula $p$
\end_inset
, they are:
\begin_inset Formula
\[
\begin{array}{c}
f(p)=\frac{n}{p}-\frac{N-n}{1-p}\\
f'(p)=-\frac{n}{^{p^{2}}}-\frac{N-n}{(1-p)^{2}}
\end{array}
\]
\end_inset
Now, the steps for Newton's method are as follows:
\end_layout
\begin_layout Enumerate
Start with a guess
\begin_inset Formula $p_{0}$
\end_inset
and compute
\begin_inset Formula $f(p_{0})$
\end_inset
and
\begin_inset Formula $f'(p_{0})$
\end_inset
.
\end_layout
\begin_layout Enumerate
Compute
\begin_inset Formula $p_{1}=p_{0}-\nicefrac{f(p_{0})}{f'(p_{0})}$
\end_inset
.
That equation can be written as
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\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $f'(p_{0})=\frac{f(p_{0})-0}{p_{1}-p_{0}}$
\end_inset
which describes a line with y values of
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\color inherit
\begin_inset Formula $f(p_{0})$
\end_inset
and 0 and the slope of the line is
\begin_inset Formula $f'(p_{0})$
\end_inset
.
\begin_inset Formula $p_{1}$
\end_inset
is the value at which the line has y-value of 0.
\end_layout
\begin_layout Enumerate
Now use
\begin_inset Formula $p_{1}$
\end_inset
in place of
\begin_inset Formula $p_{0}$
\end_inset
and repeat the calculation using the general formula
\begin_inset Formula $p_{n+1}=p_{n}-\nicefrac{f(p_{n})}{f'(p_{n})}$
\end_inset
\end_layout
\begin_layout Enumerate
Continue until
\begin_inset Formula $f(p_{n})$
\end_inset
is sufficiently close to 0 (the absolute value is less than some small