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:
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
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.
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.
An alternative route to this distribution is
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.
Mean (Submitted by Michelle Leigh)
By the Woodbury Identity it follows that
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.