On the regular conditional distribution of a multivariate Normal given a linear transformation
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.
Keywords
Cite
@article{arxiv.1612.01210,
title = {On the regular conditional distribution of a multivariate Normal given a linear transformation},
author = {Rajeshwari Majumdar and Suman Majumdar},
journal= {arXiv preprint arXiv:1612.01210},
year = {2017}
}
Comments
After this paper was uploaded in December 2016, we made substantial progress on the problem of approximating the conditional distribution given a continuously differentiable transformation. That has triggered a change in the perspective we had about this paper. As such, we are replacing it with arXiv:1710.09285