On the regular conditional distribution of a multivariate Normal given a linear transformation

TL;DR

This paper demonstrates that the conditional distribution of a multivariate Normal given a linear transformation remains Normal, using a novel operator-based decomposition approach.

Contribution

It introduces a new operator-based decomposition of multivariate Normals to prove the regular conditional distribution remains Normal.

Findings

Conditional distribution of multivariate Normal given a linear transformation is Normal

Decomposition of Normal variables into independent components using linear operators

Provides a new proof of a classical result in multivariate Normal theory

Abstract

We show that the orthogonal projection operator onto the range of the adjoint of a linear operator T can be represented as UT, where U is an invertible linear operator. Using this representation we obtain a decomposition of a multivariate Normal random variable Y as the sum of a linear transformation of Y that is independent of TY and an affine transformation of TY. We then use this decomposition to prove that the regular conditional distribution of a multivariate Normal random variable Y given a linear transformation TY is again a multivariate Normal distribution. This result is equivalent to the well-known result that given a k-dimensional component of a n-dimensional multivariate Normal random variable, where k < n, the regular conditional distribution of the remaining (n - k)-dimensional component is a (n - k)-dimensional multivariate Normal distribution.