Starting with
$ p(a) = \int p(a|b) p(b) db$
replace $p(b)$ with $\tilde{p}(b) = \mathcal{N}(b; \mu_b, \Sigma_b + \tilde{D})$
where $\tilde{D}$ is an additive diagonal covariance.
Assuming everything is Gaussian. What is mean and covariance of the resulting
$\tilde{p}(a) = \int p(a|b) \tilde{p}(b) db$ ?
My intuition is that the mean of $\tilde{p}(a)$ stays $\mu_a$ but I have trouble to spell it out especially for the resulting covariance.
