# Application in Conditional Distribution of Multivariate Normal

The Sherman-Woodbury-Morrison matrix inverse identity can be regarded as a transform between Schur complements. That is, given one can obtain by using the Woodbury matrix identity and vice versa. Recall the Woodbury Identity:

and

I recently stumbled across a neat application of this whilst deriving full conditionals for a multivariate normal. Recall that if the data are partitioned into two blocks, , then the variance of the conditional distribution is the Schur complement of the block of total variance matrix , that is, the variance of the conditional distribution is which is the variance of subtracted by something corresponding to the reduction in uncertainty about gained from the knowledge about . If, however, has the form of a Schur complement itself, then it may be possible to exploit the Woodbury identity above to considerably simplify the variance term. I came across this when I derived two very different-looking expressions for the conditional distribution and found them equivalent by the Woodbury identity. Consider the model

where

.

I was seeking the distribution and arrived there through two different paths. The distributions derived looked very different, but they turned out to be equivalent upon considering the Woodbury identity.

## Method 1

This simply manipulates properties of the multivariate normal. Marginalizing over one gets

.

Such that the distribution

It follows that the conditional distribution is

This looks a bit nasty, but notice that looks like it too could be a Schur complement of some matrix.

## Method 2

An alternative route to this distribution is

where

It follows that

which looks different from the distribution obtained through method 1. The expression for the variance is a lot neater. They are in fact identical by the Woodbury identity.

## Comparison

### Mean (Submitted by Michelle Leigh)

So mean1=mean2.

### Variance

By the Woodbury Identity it follows that

Therefore

and so variance1=variance2. The trick is recognizing the form of the formulas at the top of the page, then one can write the variance as a much neater expression.