We derive the classical result: what is the density of a multivariate normal conditioned on some proper subset of its components?

That is, if

where

then we want to characterize

# Preliminary Results

We’ll want a couple of preliminary results before establishing the primary result.

#### Marginal Distribution

Using the moment generating function, it is easy to show that the marginal distribution of $Y_1$ is

#### Block Matrix Inverse

The key piece of the derivation relies on being able to compute the inverse of a partitioned, 2x2 matrix. To establish a formula for the inverse, use gaussian elimination.

where $D = C - B^\intercal A^{-1} B$.

Hence,

# Conditional Distribution

Let $\bar{y}_1 = y_1 - \mu_1$ , $\bar{y}_2 = y_2 - \mu_2$.

Then the quadratic term in $\mathbf{Pr}(Y_1 = y_1, Y_2 = y_2)$ is

where $v^\intercal = \bar{y}_1^\intercal A^{-1} B$.

In particular, we can write the conditional density as

Equivalently,