I have a posterior probability of $p_i$ which is based on a Beta prior and some data from a binomial distribution:
I have another procedure:
$$P(E)=\prod_{i \in I} p_i^{k_i}(1-p_i)^{1-k_i}$$
which gives me the probability of a specific event of successes and failures for the set of $I$ in a model. Given the posterior distribution for $p_i$, how do I find $P(E)$?
UPDATE: I think the issue may be the notation. $P(E)$ should actually be $P(E|p_1,\dotsc,p_i,\dotsc,p_{|I|})$. Then if we are looking for the marginal probability, $P(E)$, then we need to solve for $$\int \dotsi \int_{0}^{1} P(E|p_1,\dotsc,p_i,\dotsc,p_{|I|})P(p_1,,\dotsc,p_i,\dotsc,p_{|I|})\, dp_i.$$ Because the $p_i$'s are all independent, we can probably simplify the question a lot.