# Multivariate normal with multivariate normal mean

A random vector $\boldsymbol x$ is distributed as a multivariate normal $$\boldsymbol x \sim \mathcal{N}(\boldsymbol m, \Sigma)$$ and the mean vector $\boldsymbol m$ itself is a multivariate normal $$\boldsymbol m \sim \mathcal{N}(\boldsymbol\mu, \Delta)$$

Is there a simple way to calculate the marginal probability density $$P(\boldsymbol x) = \int P(\boldsymbol x | \boldsymbol m) \; P(\boldsymbol m) \; \mathrm{d}\boldsymbol m$$ without calculating the integrals explicitly?

It should come out that $\boldsymbol x$ is distributed as $$\boldsymbol x \sim \mathcal{N}(\mu, \Sigma + \Delta)$$ but I'm having difficulties to prove it.

• Are $\Sigma$ and $\Delta$ known? Further, is this self-study or homework? – user44764 Jun 13 '14 at 17:52
• Both covariance matrices are known. This is self-study, I would like to understand how to do the marginalisation, as I know the result (just edited). – user23667 Jun 13 '14 at 17:55

## 1 Answer

Hint 1:

Rewrite $\boldsymbol x$ as $(\boldsymbol x - \boldsymbol m) + \boldsymbol m$.

Hint 2:

Consider the algebraic properties of the expectations and covariances of independent multivariate Normals.

• Thank you, that was exactly what I needed. I thought there was an easy way out of this, and it helped me solve my actual problem that lead to this question. – user23667 Jun 13 '14 at 19:56