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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...

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According to the abstract of this paper,

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.

However, I imagine the closed form requires a great deal of combinatorial calculation and hence would not be practically useful. The slow numerical algorithm you mentioned is probably faster.

This paper may be more useful since it does not require integer parameters.

The distribution of product of independent beta random variables with application to multivariate analysis

I haven't read the paper, but the abstract sounds promising.

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    $\begingroup$ For record keeping: The DOI of the "The distribution of product of independent beta random variables with application to multivariate analysis" is 10.1007/BF02480942. $\endgroup$ – fuenfundachtzig Oct 6 '10 at 13:59
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    $\begingroup$ The other one is 10.1137/0118065 :) $\endgroup$ – fuenfundachtzig Oct 6 '10 at 14:00

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