Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Marginal distributions

Theorem: Let $x$ follow a multivariate normal distribution:

$\label{eq:mvn} x \sim \mathcal{N}(\mu, \Sigma) \; .$

Then, the marginal distribution of any subset vector $x_s$ is also a multivariate normal distribution

$\label{eq:mvn-marg} x_s \sim \mathcal{N}(\mu_s, \Sigma_s)$

where $\mu_s$ drops the irrelevant variables (the ones not in the subset, i.e. marginalized out) from the mean vector $\mu$ and $\Sigma_s$ drops the corresponding rows and columns from the covariance matrix $\Sigma$.

Proof: Define an $m \times n$ subset matrix $S$ such that $s_{ij} = 1$, if the $j$-th element in $x_s$ corresponds to the $i$-th element in $x$, and $s_{ij} = 0$ otherwise. Then,

$\label{eq:xs} x_s = S x$

and we can apply the linear transformation theorem to give

$\label{eq:mvn-marg-qed} x_s \sim \mathcal{N}(S \mu, S \Sigma S^\mathrm{T}) \; .$

Finally, we see that $S \mu = \mu_s$ and $S \Sigma S^\mathrm{T} = \Sigma_s$.

Sources:

Metadata: ID: P35 | shortcut: mvn-marg | author: JoramSoch | date: 2020-01-29, 15:12.