On the conditional distribution of a multivariate Normal given a transformation – the linear case

Abstract

We show that the orthogonal projection operator onto the range of the adjoint T⁎ of a linear operator T can be represented as UT, where U is an invertible linear operator. Given a Normal random vector Y and a linear operator T, we use this representation to obtain a linear operator Tˆ such that TˆY is independent of TY and Y−TˆY is an affine function of TY. We then use this decomposition to prove that the conditional distribution of a Normal random vector Y given TY, where T is a linear transformation, 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 Normal random vector, where k<n, the conditional distribution of the remaining (n−k)-dimensional component is a (n−k)-dimensional multivariate Normal distribution, and sets the stage for approximating the conditional distribution of Y given g(Y), where g is a continuously differentiable vector field.