Revision as of 08:58, 5 April 2014

Tutorial on Maximum Likelihood Estimation: A Parametric Density Estimation Method

MLE Tutorial in PDF Format

Motivation

Suppose one wishes to determine just how biased an unfair coin is. Call the probability of
tossing a HEAD is p. The goal then is to determine p.

Also suppose the coin is tossed 80 times: i.e., the sample might be something like x₁ = H,
x₂ = T, …, x₈ = T, and the count of number of HEADS, "H" is observed.

The probability of tossing TAILS is 1 − p. Suppose the outcome is 49 HEADS and 31 TAILS,
and suppose the coin was taken from a box containing three coins: one which gives HEADS
with probability p = 1 / 3, one which gives HEADS with probability p = 1 / 2 and another which
gives HEADS with probability p = 2 / 3. The coins have lost their labels, so which one it was is
unknown. Clearly the probability mass function for this experiment is binomial distribution with
sample size equal to 80, number of successes equal to 49 but different values of p. We have
the following probability mass functions for each of the above mentioned cases:

$ Pr(H = 49 | p = {1}/{3}) = \binom{80}{49}(1/3)^{49}(1 - 1/3)^31 \approx 0.000 $

$ Pr(H = 49 | p = {1}/{2}) = \binom{80}{49}(1/2)^{49}(1 - 1/2)^31 \approx 0.012 $

$ Pr(H = 49 | p = {2}/{3}) = \binom{80}{49}(2/3)^{49}(1 - 2/3)^31 \approx 0.054 $

Based on the above equations, we can conclude that the coin with p = 2 / 3 was more likely
to be picked up for the observations which we were given to begin with.

Definition

The generic situation is that we observe a n-dimensional random vector X with probability
density (or mass) function f(x / θ). It is assumed that θ is a fixed, unknown constant
belonging to the set $ \Theta \subset \mathbb{R}^{n} $.

For $ x \in \mathbb{R}^{n} $, the likelihood function of θ is defined as 

L(θ / x) = f(x / θ)

<span class="texhtml" />x is regarded as fixed, and θ is regarded as the variable for L. The log-likelihood function
is defined as l(θ / x) = l'o'g'L(θ / x).

The Maximum Likelihood Estimate (or MLE) is the value $ \hat{ \theta } = \hat{\theta(x)} \in \Theta $
maximizing L(θ / x), provided it exists:

$ L(\hat{\theta}/(x)) = \underset{\theta}{argmax}[ L(\theta/x) ] $

What is Likelihood function ?

If the probability of an event X dependent on model parameters p is written as
P(X | p)