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

That is, if


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 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 .


Conditional Distribution

Let , .

Then the quadratic term in is

where .

In particular, we can write the conditional density as