I have a $D$ dimensional distribution $q$ which can be written as proportional to the following-
$q(z) \propto \prod_{i=1}^D \mathcal{N} \big(z_{i}| \eta_i, \lambda_i^{-1} \big) \times \mathcal{N}(z| \mu, \Sigma)$
where the $\eta_i$ and $\lambda_i^{-1}$ are depend on $i$. Notation: $\lambda_i^{-1}$ is the variance.
How should I simplify this? I know that a product of Gaussians will be a Gaussian hence the proportionality constant will go away since the result can be normalized.
I tried the following -
$$ \begin{align} \ln q&(z) = \sum_{i=1}^D \ln \mathcal{N} \big(z_{i}| \mu_i, \lambda_i^{-1} \big) + \ln \mathcal{N}(z| \mu, \Sigma) + \text{const} \\ &= - \frac{1}{2} \sum_{i=1}^D \lambda_i \big(z_{i} - \mu_i \big)^2 - \frac{1}{2} (z -\mu)^T\Sigma^{-1} (z-\mu) + \text{const} \,. \nonumber \end{align} $$
but I don't know how to proceed further. Can someone please help me derive a concise expression for the distribution $q(z)$ which is simply just a Multivariate Gaussian?
