What I have are n different time series of bernouli processes of varying lengths, taking the values of 0 or 1. What I would like to do is to use Bayesian inference to calculate, for one of these series, the posterior distribution of the parameter p that drives the proccess. To do this I first need the prior distribution of the Bernouli paramter p which I gather should be available to me through all the n time series that I have. But how do I do this in practice?
To find the mean of the prior distribution of p seems straight forward enough, just count the total number of occurances in all times series and divide by the total number of trials. The variance on the other hand seems a bit less clear to me. I imagine I could take sample means (i.e. frequentist maximum likelihood estimator of p) from each of the n processes and then compute a sampl variance from all of these, is that appropriate? Intuitively it feels not quite right to me since these estimators will themselves not be very indicative of the true value at all (this is in fact the whole reason I am trying to use Bayesian inference here) since they will likely not the long enough. For example I may have one of the n processes be of length ~50 with the true p being something like 0.2.
Is there any way to adress this issue or is the approach I mentioned the best we can accomplish? Will the prior value of the variance affect the results I get in my Bayesian inference later a lot, or is it not so impactful?
Edit: Adding my comment to Tim's answer here to clarify the goal:
I think I should clarify that I do not postulate that all of these Bernoulli Processes have the same p. In fact, I am assuming that they have different values of p. My goal is to find the distribution of p for one specific process (to make things clear we can say that the data available from this process was not used to find the prior). To this end, I need a prior distribution of p, and I'm thinking this one should be found via the other available processes. Does my question make more sense framed like this?
We could for example say that the possible values of p are 0.3, 0.5, and 0.7 all with the same probabilities. If this were the case then we would use that as the prior and then after some observations become more confident that we are dealing with one of them rather than the others. So it is this (unknown) prior distribution that I need to find