Given two multivariate gaussians distributions, given by mean and covariance, $G_1(x; \mu_1,\Sigma_1)$ and $G_2(x; \mu_2,\Sigma_2)$, what are the formulae to find the product i.e. $p_{G_1}(x) p_{G_2}(x)$ ?
And if one was looking to implement this in c++, what would an efficient way of doing it?
Go easy, I am primarily a computer scientist and not a pure mathematician.
Any help much appreciated.
Best Answer
An alternative expression of the PDF proportional to the product is:
$\Sigma_3 = \Sigma_1(\Sigma_1 + \Sigma_2)^{-1}\Sigma_2$
$\mu_3 = \Sigma_2(\Sigma_1 + \Sigma_2)^{-1}\mu_1 + \Sigma_1(\Sigma_1 + \Sigma_2)^{-1}\mu_2$
The advantage of this form for computation is that it requires only one matrix inverse.