# How to use the estimate given by the EM algorithm to guess at the missing value

Text: Computational Statistics 2E by Givens and Hoeting

Example 4.1: Simple Exponential Density

The set-up is as follows: Suppose that $Y_1, Y_2 \overset{\text{iid}}{\sim} \rm{Exp}(\theta)$ and that we observe $y_1 = 5$ and $y_2$ is missing.

E-step: $Q(\theta | \theta^{(t)}) = 2 \log \theta - 5 \theta - \theta y_2$

M-step: $\theta^{(t+1)} = \frac{2 \theta^{(t)}}{5 \theta^{(t)} + 1}$

The authors state that using an initial value the estimate will converge to $\hat{\theta} = 0.2$. I have verified this using R using an initial value as 1. The authors also state that the EM algorithm is not necessary in finding the MLE for $\theta$, which in this case would be $\bar{y} = \frac{y_1 + y_2}{2}$.

My question is as follows: Suppose we have the same setup as above, only now we are interested in guessing the value of $y_2$ when $y_1 = 5$. How would we go about providing an "educated" guess for $y_2$?

My attempt: Since we know the MLE exactly, we could then use the value we obtained via the EM algorithm and solve for $y_2$. By that I mean,

$\frac{y_1 + y_2}{2} = \frac{5 + y_2}{2}= 0.2 \Rightarrow y_2 = -4.6$