If we know that a,b are both Gaussian distributions, then we have that:
\begin{equation} \mqty[a \\ b] \sim \mathcal{N} \left(\mqty[\mu_{a} \\mu_{b}], \mqty[A & C \\ C^{T} & B]\right) \end{equation}
whereby: A is the covariance of each element of A B is the covariance of each element of B C is the covariance of A against B To perform inference:
\begin{equation} p(a|b) = \mathcal{N}(a | \mu_{a|B}, \Sigma_{a|b}) \end{equation}
wherby:
\begin{equation} \mu_{a|b} = \mu_{a} + CB^{-1}(b-\mu_{b}) \end{equation}
\begin{equation} \Sigma_{a|b} = A - CB^{-1}C^{T} \end{equation}
Its a closed form solution. Tada. We know that B is positive semidefinite, and that its invertible, from the fact that its a covariance.