Having bayesian estimates of a proportion is relatively easy. You model that proportion as a binomial variable, you choose a beta-binomial prior and by using the likelihood you obtain a beta-binomial posterior.
I have problems providing bayesian estimates of a weighted proportion. That is, I wont to compute estimates of the variable: $$p' = (\sum w*success)/\sum w $$ instead of $$p = (\sum success)/n$$ I don't even know if I need to provide a distribution of the weighting variable, but if I had to I would choose a normal distribution.
Is there a way to use bayesian inference to estimate the distribution of p'?