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