# 2.3.4 - Maximum Likelihood Estimation

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

$$\pi_1+\ldots+\pi_k=1$$

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}

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