On the conditional distribution of a multivariate Normal given a transformation – the linear case
Rajeshwari Majumdar, Suman Majumdar · Heliyon · 2019
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 T ˆ such that T ˆ Y is independent of TY and Y − T ˆ Y is an affine function of TY . We then use this decomposition to prove that the conditional distribution of a Normal random vector Y given T Y , 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.