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
We’ll want a couple of preliminary results before establishing the primary result.
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.
Let , .
Then the quadratic term in is
In particular, we can write the conditional density as