2.3.4 - Maximum Likelihood Estimation

If \(X \sim Mult\left(n, \pi\right)\) and we observe \(X = x\), then the loglikelihood function for \(\pi\) is

\(L(\pi)=\log\dfrac{n!}{n_1!\cdots n_k!}+x_1 \log\pi_1+\cdots+x_k \log\pi_k\) 

We usually ignore the leading factorial coefficient because it doesn't involve \(\pi\) and will not influence the point where \(L\) is maximized. Using multivariate calculus with the constraint that


the maximum is achieved at the vector of sample proportions:

\begin{align} \hat{\pi} = \dfrac{1}{n}x= (x_1/n,x_2/n,\ldots,x_k/n)\\ \end{align}