Exercise 3

EM for multinomial

Published

October 18, 2024

This is one of the original illustrating examples for the use of the EM algorithm.

Suppose that a vector of observed counts \(y=(125,18,20,34)\) arises from a multinomial distribution with cell-probabilities \((\frac{1}{2}+\frac{\theta}{4}, \frac{1-\theta}{4}, \frac{1-\theta}{4}, \frac{\theta}{4})\). The aim is to find the ML estimate of \(\theta\).

The density of the observed data is \[ f_{Y \mid \Theta}(y \mid \theta)=\frac{n !}{y_{1} ! y_{2} ! y_{3} ! y_{4} !}\left(\frac{1}{2}+\frac{\theta}{4}\right)^{y_1}\left(\frac{1}{4}-\frac{\theta}{4}\right)^{y_2}\left(\frac{1}{4}-\frac{\theta}{4}\right)^{y_3}\left(\frac{\theta}{4}\right)^{y_4} \] Although the log-likelihood can be maximized explicitly, we use the example to illustrate the EM algorithm.

To view the problem as an unobserved data problem, we would think of it as a multinomial experiment with five categories with observations \(x=\left(y_{11}, y_{12}, y_2, y_3, y_4, y_5\right)\), each with cell probability \((1 / 2, \theta / 4,(1-\theta) / 4,(1-\theta) / 4, \theta / 4)\). That is, we split the first category into two and we can only observe the sum \(y_1=y_{11}+y_{12}\).