# Product of beta distributions

I am looking at trigger efficiencies, meaning that I have some device that fires on $k$ out of $n$ events. In the end I am interested in some estimate of the efficiency $\epsilon$ which is the probability to fire on a randomly given event. Using a Bayesian approach with a uniform prior over $[0,1]$ I can model the probability distribution for $\epsilon$ as a Beta distribution $\beta(\epsilon; k+1, n-k+1)$.

Now comes the question: I calculate the efficiency using "bootstrapping" which means that the final trigger efficiency is the product of two trigger efficiencies, both of which can be modelled as Beta distributions.

How can I calculate this product of the two Beta PDFs for large values of $k_{1,2}$ and $n_{1,2}$ efficiently? Is there a closed form of the product (AFAIK not)? At the moment I am doing this numerically, but this is rather slow.

This question has the answer how to evaluate integrals of the Beta distribution for large argument values, but this does not help here.

I hope my question is clear and not completely stupid...

The density function of products of random beta variables is a Meijer $G$-function which is expressible in closed form when the parameters are integers.