# How to estimate the likelihood function for random generator of three events?

I have a random events generator. I know in advance the set of event that can be generated (in my case I have only three possible events). The probabilities of the events are not known. I need to estimate these probabilities. For that I run an experiment. For example I generate 20 events. So, I have a sequence of events. For example: a,a,b,a,c,c,c,a,b,....c. Having the sequence I can count for the number of every event (so, in my case I get three integer numbers $n_1$, $n_2$ and $n_3$).

I can calculate the probabilities of every event in the following way:

$\nu_1 = \frac{n_1}{n_1+n_2+n_3}$ $\nu_2 = \frac{n_2}{n_1+n_2+n_3}$ $\nu_3 = \frac{n_3}{n_1+n_2+n_3}$

But these values are approximate. For example, I can have probability of 0.25 for the event a, so if I generate 20 events, I have to get approximately 5 a-events. But just by chance I can get 10 or 0 a-events.

So, I want to have a density distribution of probabilities. Since, $\nu_1$, $\nu_2$ and $\nu_3$ are dependent (the sum of them is equal to one) I going to use $\nu_1$ and $\nu_2$. So, I want to have an explicit form for the density distribution of $\nu_1$ and $\nu_2$.

$\rho (\nu_1,\nu_2) = F (n_1,n_2,n_3)$.

Does anybody know where I can get it?

• The terminology you are using in your question is a little bit nonstandard: for example, your title asks for a "likelihood" but your problem description asks for a probability distribution on $(\nu_{1},\nu_{2})$. In short, a Bayesian solution would start with a (conjugate) Dirichlet prior on $(\nu_{1},\nu_{2})$, then you would observe a multinomial likelihood with $(n_{1},n_{2})$, then the posterior would be Dirichlet again, equivalent to your $F(n_{1},n_{2},n_{3})$, above. This gives you a density, but it isn't entirely clear from your description that this is what you want.
– user1108
Commented Dec 17, 2010 at 1:59
• By the way, the solution outlined above is written out in more detail at the bottom of this other problem: stats.stackexchange.com/questions/3194/…
– user1108
Commented Dec 17, 2010 at 2:04

## 1 Answer

You should look at the multinomial distribution.