= Maximum Likelihood =

'''Maximum likelihood''' is a method for fitting a distribution to observations.

<<TableOfContents>>

----



== Description ==

The most common probability distribution functions are parameterized so that shape, scale, etc., can vary to better fit a particular set of observations. A methodology is then needed to determine the values for those parameters with the 'best' fit.

For example, the probability density function for an [[Analysis/ExponentialDistribution|exponential distribution]] with a rate of ''λ=λ*'' is:

{{attachment:pdf.svg}}

The '''likelihood''' for a value of ''λ'' given an observation ''x,,i,,'' is therefore given by:

{{attachment:lik1.svg}}

Given many observations ''i'' in ''X'', the probabilities are multiplied.

{{attachment:lik2.svg}}

The method of '''maximum likelihood estimation''' relies on [[Calculus/CriticalPoint#Logarithmic_differentiation|logarithmic differentiation]] and more generally on logarithm properties. The log of this product is taken, turning the expression into a summation that usually can be simplified, and that is still useful for optimization.

{{attachment:lik3.svg}}

Let ''n'' be the number of observations in ''X'' so that:

{{attachment:lik4.svg}}

The value of ''λ'' that maximizes likelihood is a  [[Calculus/CriticalPoint|critical point]], i.e. identified by deriving the expression and setting to zero.

{{attachment:lik5.svg}}

{{attachment:lik6.svg}}

{{attachment:lik7.svg}}

{{attachment:lik8.svg}}

So the maximum likelihood estimator (MLE) for ''λ'' is ''λ,,MLE,, = 1/X̅''.



----
CategoryRicottone