I am trying to reconcile what I think are conflicting philosophies in my head about an experiment.
The experiment is a hierarchical model where:
X ~ Bernoulli(p)
p has an exact functional relationship of the form:
p = p(theta, B, f).
B is a parameter that is fixed between all realizations of an ideal experiment, but unknown, and we want to infer it using Markov Chain Monte Carlo techniques. "f" is a fixed, known number for the entire experiment.
In each realization of an ideal experiment, however, theta is random and distributed with PDF sin(theta)/2.
What is measured in the practical experiment is an *average* of these repetitions, i.e. total number of successes, i.e. so we measure a number that is the fraction of successes over ~10^8 realizations of the experiment.
The problem I am having is that I initially thought I would assume a prior distribution of sin(theta)/2 on theta and then just do MCMC to infer the other parameters, but this is illogical and is really the assumption that theta is *fixed* and our prior knowledge follows a particular distribution, which should get much tighter as the repetitions of the experiment reveal the "true" value of theta in the posterior distribution.
In fact, though, I want to know the average p*n value, translate that into some sort of Poisson distribution, and then apply MCMC to infer B and marginalize out everything else.
In truth, the only way I can imagine this experiment being setup in a fully Bayesian way is that for each realization of the experiment, we multiply by the likelihood and prior for theta_i, where i runs from 1 to N = 10^8, since theta_i has a definite value in the experiment but we just don't know it, and I want to assume that it's drawn from a distribution with PDF sin(theta)/2. Then perhaps there is some clever way to marginalize over all theta_i values and come up with an answer that is equivalent to just doing some sort of distribution transformation (integration).
Can someone clear up the fog I am in about how to properly think about/treat this situation? Thank you so much.