Marginalization from a distribution

I have a posterior distribution over a set of parameters denoted by $\theta = \{w, \phi, \lambda\}$ and the posterior is the joint distribution given the observed data i.e. $P(\theta | D)$. Now, this posterior is a multivariate normal distribution (due to the approximate way it is estimated).

Now I want to find out the marginal distributions from this joint. So, for example, $P(\lambda)$ could be found my marginalizing over $w$ and $\phi$. Is there an easy way to do this for such MVN distributions? Also, can I find the mode for this marginal distribution easily?

• Luca, if these are still the distributions you were discussing in your previous question, the independence of $w,\phi$ and $\lambda$ gives you the marginals immediately - the $q(w)$, $q(\phi)$ and $q(\lambda)$ you had in the previous question. Commented Feb 11, 2014 at 11:37
• Yes, they are sort of related. I have so many doubts, I am trying to figure them out one by one!
– Luca
Commented Feb 11, 2014 at 11:58

$$p(\theta|D) = Normal(\mu, \sigma)$$
with $\mu$ a 3 dimensional vector (position 1 corresponding to $w$, position 2 corresponding to $\phi$ and position 3 corresponding to $\lambda$) and similarly for the 3x3 covariance matrix.
Then the marginal $p(\lambda|D) = Normal(\mu_3, \sigma_{33})$. You simply pick up the corresponding position from the mean vector and covariance matrices.