I have a dataset of $m$ individuals. For each individual $m$ I have $n_m$ (binomial ) observations with $s_m$ corresponding to the number of 'successes'.
I use this data to fit a beta-binomial Bayesian model (with a conjugate prior) and update it likewise. This way I can estimate $p_m$ (probability of success) for a (possible new) individual given the observations (if existing) on that (new) individual.
However, on many individuals in my dataset I have only very few observations. These are uncertain records in my dataset. However, there is a relation between the number of observations and $p_m$. So, I should somehow also take into account these 'uncertain individuals' in my dataset. In other words: individuals with a low number of observations tend to have lower $p_m$'s (which I have proven to be true for my specific dataset). If I cut those individuals out (in order to get a more certain dataset) I am getting biased outcomes.
I am thinking about using either Beta-Binomial regression, or fitting in a Poisson-Gamma model as well. However, $n_m$ (number of observations on individual $m$) can grow towards infinity, and, well, I don't know if this is the best way to go? (Overdispersion?)
I don't have much experience with these kinds of models, so I would like to ask what could be a good way forward (Beta-Binomial regression?), and if someone could explain this using my example in simple terms? Thank you!