*I did not really know how to choose an adequate title for this question, so please feel free to change it.*

I have a weird case wherein frequentist and Bayesian philosophies come together. I am dealing with a data set of observations on persons who were faced with a situation in which they could either perform an action or could decide not to. The data set consists of several persons with a varying number of observations on that person.

In a frequentist approach, I could say that the probability of a certain person performing that action is the number of times the person performed the action given the situation divided by the number of times the situation occurred. So: if the situation in which the person *could* perform the action occured $23$ times, and this person has performed the action $8$ times, we could say the probability of this person performing this action in that situation is $\frac{8}{23} = .348$.

Now, when I encounter a *new* person (not in the data set I already have), I want to have a reasonable *estimate* of the probability of this person performing this action in the said situation, given very few or no observations at all.

What I do (and what works) is:

 - I consider my data set of historical observations and I **leave out all persons with less than $x$ observations** (in my case: 100 observations).
 - I fit a mixture of gamma distributions to my data set (since there are different clusters of persons in my data set) using EM.
 - I use Bayes' theorem (with a conjugate prior, my fitted Beta distribution is the prior) and update using Bayesian inference to get my *probability estimate* (and corresponding credible interval) for the newly observed person (of this person performing this action in the said situation).

However, I do not like that I am cutting out all persons with a lower than $x$ number of observations in my prior data set. In my particular case, persons with lower amounts of observations tend to belong to a different cluster compared to persons with a very high amount of observations. So: there is a correlation between the amount of observations and the probability I am trying to calculate! I feel like my prior is biased if I cut out some people (even if they have very few observations and are thus very unreliable).

My question: is there any alternative to my method of cutting out persons with observations below a certain number $x$? Can I maybe somehow still take these persons into account and define their (frequentist) probabilities relative to each other? Can I weigh them into the prior Beta distribution using their credible intervals calculated relative to the other players in the data set? Maybe I should use some sort of regression and factor in another variable for the number of observations..? Or is any other method than the method I am using just *impossible*?