Let $X$ be a $d$-dimensional random vector drawn from a Gaussian mixture
$$ X \sim \sum_{k=1}^K \pi_k \, \mathcal{N}_d(\mu_k, \Sigma_k), $$ and let
$$ Y = X + N, \quad N \sim \mathcal{N}_d(0, \Sigma_N), $$ with $N$ independent of $X$.
The MMSE estimator is
$$ \hat{X}(Y) = \mathbb{E}[X \mid Y], $$ and I want the expected squared error
$$ \mathbb{E}\!\left[\|X - \hat{X}(Y)\|^2\right]. $$
I can derive this quantity directly (it involves the posterior mixture weights and Gaussian integrals), but the expression is long and not central to my main topic. I’m writing a scientific paper where this is only a technical intermediate result, so I’d prefer to cite a standard reference, i.e., a book or article that explicitly presents or derives the MMSE (expected error) for a Gaussian mixture prior with additive Gaussian noise. It would also save me greatly on the space in the submission, which is of course limited.
Could anyone point me to such a reference in estimation theory, Bayesian statistics, or signal processing?
