Example 7.2.19 from Casella & Berger (EM algorithm)

Question

$X_1, ..., X_n \sim Poi(\tau_i), Y_1, ..., Y_n \sim Poi(\beta\tau_i).$ $X$ and $Y $are mutually independent. However, $X_1$ is missing. I would like to use the EM algorithm to estimate $\beta$ and $\tau_i$. The expected complete-data log likelihood is shown on the top of page 329 in the red box

I do not understand why $\beta^{(r+1)} = \frac{\sum_{i=1}^{n}y_i}{\tau_1^{(r)}+\sum_{i=2}^{n}x_i}$. I differentiated the expected complete-data log likelihood function and got $\beta^{(r+1)} = \frac{\sum_{i=1}^{n}y_i}{\sum_{i=1}^{n}\tau_i}$. I am pretty new to EM algorithm and I don't known where I got wrong. Thanks a lot to anyone who can give me a hint.

Answer 1

Actually $\tau_1$ needs to be differentiated separately to find its MLE estimator for $r+1$.

Set the derivatives to 0 and solve to get the MLEs for $\tau_1$ and $\tau_j, j>=2$. These should come out the same as the authors.

Then substitute these values into your solution for $\beta^(r+1)$ and isolate $\beta^(r+1)$ again and you will end up with the same result as the authors.

There is no typo. (Actually there is a different typo in older editions at the top of page 329--which is why I came to this page, but your edition has this corrected).

Answer 2

I am trying to solve this EM problem, and I got the exactly same answer as yours. So, I believe we are correct and the answer in the book is a typo. My answer is

$\beta^{(r+1)} = \frac{\Sigma_{i=1}^{n}y_i}{\Sigma_{i=1}^{n}\tau_i^{(r)}}$

$\tau_1^{(r+1)} = \frac{y_1+\tau_1^{(r)}}{1+\beta^{(r+1)}}$

$\tau_i^{(r+1)} = \frac{y_i+x_i}{1+\beta^{(r+1)} }$ for $i=2,3,\ldots,n$

Answer 3

You're on the right track. In the full likelihood function (7.2.11), after taking derivative on all the $\tau_i$'s, you'll find that $\Sigma_{i=1}^{n}\tau_i = \frac{\Sigma_{i=1}^{n}{(x_i+y_i)}}{\beta+1}$. Replace $\Sigma_{i=1}^{n}\tau_i$ in your results with $\frac{\Sigma_{i=1}^{n}{(x_i+y_i)}}{\beta+1}$ and you'll get the same results as the authors.