Suppose you have $\theta=\{1,2\}$ and the sample of (0,1,2) with the task of finding MLE:
\begin{array} {|c|c|c|} \hline x & p(x|\theta=1) & p(x|\theta=2) \\ \hline 0 & 1/2 & 1/4 \\ \hline 1 & 0 & 1/4 \\ \hline 2 & 1/3 & 0 \\ \hline \end{array}
I can't compare $\prod_i p(x_i|\theta)$, because of the zeroes. Even though I can compare probability for a single observation, I'm still having trouble quantifying the difference. $p(x=1|\theta=2)>p(x=1|\theta=1)$ for x=1, but can I do a similar comparison for the whole vector?
Edit
The original question is
One observation is taken on a discrete random variable X with pmf f(x|θ), where θ∈{1,2,3}. Find the MLE of θ.
This is followed by a 4x5 table which includes zeroes in every column. I think the idea behind this question is that we have a distribution that doesn't describe population well. Do we just say that the method of MLE is not applicable here?