Let's say that I have a mixture of Gaussians representing a likelihood:
$$ p(\mathbf{x}) = \sum_{i=1}^K\phi_i \mathcal{N}(\boldsymbol{\mu_i,\Sigma_i}) $$
What is the posterior distribution given a prior Gaussian distribution $\mathcal{N}(\boldsymbol{\mu_p,\Sigma_p})$?
Is it simply this? $$ p(\mathbf{x}) = \sum_{i=1}^K\phi_i \mathcal{N}(\boldsymbol{\mu_i,\Sigma_i})\mathcal{N}(\boldsymbol{\mu_p,\Sigma_p})\\ = \sum_{i=1}^K\phi_i \mathcal{N}(\boldsymbol{\mu_{i,p},\Sigma_{i,p}}) $$ with:
$$ \Sigma_{i,p} = \Sigma_i(\Sigma_i + \Sigma_p)^{-1}\Sigma_p\\ \mu_{i,p} = \Sigma_p(\Sigma_i + \Sigma_p)^{-1}\mu_i + \Sigma_i(\Sigma_i + \Sigma_p)^{-1}\mu_p $$