Maximum likelihood is a method for estimating the parameters of a distribution when the form of the
distribution is already known and a dataset that has been generated from the distribution is available.
The distribution is assumed to be dependent on a vector of parameters, {% \vec{\theta} %}.
The method seeks to identify the most likely value of {% \vec{\theta} %} given a
dataset {% (x_1,x_2, ... , x_n) %}
The likelihood function is defined as the value of the density function for the given dataset, given a specified value
for the parameter vector.
{% likelihood(\theta) = f(x_1,x_2, ... , x_n|\theta) %}
Once this function is defined, the method of maximum likelihood seeks to find the maximum value of this function
by varying the parameter vector.
{% likelihood(\hat{\theta}) = max _{\theta} likelihood(\theta) %}
In general, the method usually proceeds to find the maximum of the logarithm of the likelihood instead of the
likelihood. This is because finding the max of the log likelihood is generally easier to do, and because
the logarithm is a monotonically increasing function, the maximum of the log likelihood is also the
maximum of the likelihood.
{% l(\theta) = \sum log f(x_{\alpha}|\theta) %}
