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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.

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  • $\begingroup$ Are $\Sigma$ and $\Delta$ known? Further, is this self-study or homework? $\endgroup$
    – user44764
    Commented Jun 13, 2014 at 17:52
  • $\begingroup$ 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). $\endgroup$
    – user23667
    Commented Jun 13, 2014 at 17:55

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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.

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  • $\begingroup$ 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. $\endgroup$
    – user23667
    Commented Jun 13, 2014 at 19:56

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