I have a candidate C with unknown features $F = [F_1, F_2, … , F_n]$
where $F_i$ is the probability that the candidate has the feature , unfortunately this is poorly know and must be approximated so that $F_i$ ~ $N(\mu_{fi},\sigma_{fi}^2)$
I appreciate that this is not bounded between 0 and 1 but the Gaussian approximation is useful for improving the estimate as more data is gathered.
There is also a user with likes $[U_1, U_2, …, U_n]$, again this is approximated $U_i$ ~ $N(\mu_{ui},\sigma_{ui}^2)$
Hence the probability that the candidate will have at least one feature that the user likes is the 1 minus the probability that the venue has no features that the user likes. This is calculated from the product of the probabilities that not(has feature and user likes feature) for each feature: $$P(C) = 1 - P(\bar C) = 1 - \Pi (1- F_i U_i ) $$
I believe that the mean of this distribution is: $$\mu_c = 1 - \Pi(1- \mu_{fi} \mu_{ui}))$$
And I have been able to confirm through simulation that the distribution is a gaussian (or very well approximated by gaussian), but what's it's variance? Any help would be much appreciated, I'm worried I may have to perform a tricky integral and have no idea how, n is roughly 370 so ideally I'm looking for a format which scales in a programable way. Thanks in advance
I have a solution to my issue - the product of two gaussians is not a Gaussian as mentioned before, but the result can be approximated as a Gaussian with $mu = \mu_1*\mu_2$ and $\sigma^2 = \sigma_1^2 *\sigma_2^2 + \sigma_1^2*\mu_1^2 + \sigma_2^2 * \mu_2^2$
This is the result from this paper also referenced on mathoverflow. This approximation gets better as either $\mu_i/\sigma_i $ gets large, but is surprisingly good as long as either $\mu_i$ is non zero (anecdotally)
If the true distribution is needed then the output can be found by looking at the Normal Product Distrubution.