If we apply Bayesian inference to try and determine the distribution of a multivariate Gaussian $\textbf{x}$, and we have two predictions
$$ \textbf{x}\sim N(\textbf{a}_1,\Sigma _1)~~ and ~~ \textbf{x}\sim N(\textbf{a}_2,\Sigma _2) $$ based on different data sets then is there a way of combining these into a single refined estimate of the distribution of $\textbf{x}$? I want to just multiply the two Gaussians since this gives a Gaussian, but the result in un-normalised and so isn't a PDF. Can we simply normalise the product?
Take a covariance weighted average.
(Also, your notation here needs fixing: you need to define the x as samples from a distribution)