On the Conditional Distribution of a Multivariate Normal given a Transformation - the Linear Case

TL;DR

This paper proves that the conditional distribution of a multivariate Normal vector, given a linear transformation, remains Normal, by decomposing the vector into independent components and leveraging properties of linear operators.

Contribution

It introduces a novel decomposition of Normal vectors using linear operators, providing a new proof that their conditional distributions given linear transformations are also Normal.

Findings

Conditional distribution of a Normal vector given a linear transformation is Normal.

Decomposition of Normal vectors into independent components using linear operators.

Framework for approximating conditional distributions given nonlinear functions.

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 Normal random vector $Y$ 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 conditional distribution of a Normal random vector $Y$ given a linear transformation $\mathcal{T}Y$ 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 $\left(n-k\right)$-dimensional component is a $\left(n-k\right)$-dimensional multivariate Normal distribution, and sets the stage for approximating the conditional distribution of $Y$ given $g\left(Y\right)$, where $g$ is a continuously differentiable vector field.