I am trying to solve a problem and the results I get seem counter-intuitive.
We randomly throw $n$ balls into an area partitioned into 3 bins $b_1,b_2,b_3$. The size of each bins is proportional to the probability the ball will fall in it. Let's call these probabilities $p_1,p_2,p_3$. This can be described by a multinomial distribution.
Now let's say I throw 12 balls, and I know how many landed in each bin ($x_1=3,x_2=6,x_3=3$).
I would like to estimate the size of the bin from the observations. For this I use Maximum Likelihood. It can be shown that the MLE will be $p_1=3/12,p_2=6/12,p_3=3/12$. This is pretty intuitive.
It turns out that the actual likelihood at this point is:
Now, let's assume I knew in advance that $p_1=p_3$. How would that change my result? It would not - I would still get the same parameter values $p_1=0.25,p_2=0.5,p_3=0.25$.
The twist comes now: let's assume I cannot observe balls that landed in $b_3$. If I know that 12 balls were thrown I am fine, since I can calculate $b_3=n-b_1-b_2=12-3-6=3$. but what happens if I don't know $n$?
I figure that in this case, I would need to estimate $x_3$ (or equivalently $n$) as well. However, if I use MLE, the results start looking weird. Intuitively, I would expect that if I observe $x_1=3,x_2=6$ and I know that $p_1=p_3$, then the MLE will probably be $p_1=0.25,p_2=0.5,p_3=0.25,x_3=3$. However, it is clearly not the maximum, since for example:
So from this it seems that $x_1=3,x_2=6,x_3=2$ is more likely than $x_1=3,x_2=6,x_3=3$ even if I know that $p_1=p_3$, which seems very counter-intuitive.
My questions are whether my logic is sound, whether my intuition is misleading me and whether this is the correct way to estimate the parameters and missing data.
For future reference, I found this highly relevant paper, which addresses this exact problem. It also discusses the slight skew mentioned in whuber's answer.