Hi, given a D dimensional multivariate Gaussian distribution and an arbitrary hyperplane in this D dimensional space, is it possible to compute the conditional Gaussian distribution on the hyperplane?

I know that if the hyperplane's axes corresponds to a subset of the axes of the D dimensional space, then this is can be computed using the standard formula for conditioning a Gaussian distribution.

However if the hyperplane is arbitrarily oriented then it seems to suggest that the original Gaussian distribution need to be rotated (so the axes matches the hyperplane's axes) and then apply the standard formula. But I can't quite figure out how to generate this rotation matrix from just the hyperplane informations.

Can anyone help me? Or suggest a better way?

thanks

Shaobo

Actually, the hyperplane can have dimensionality of D-K, where K < D.

What do you mean by "direction cosines of the arbitrary vector"? And assume that even though the other rows are arbitrary, they still needed to be generated in order to create the D x D orthonormal matrix, how is this done?

thanks

Shaobo

How are you specifying your hyperplane?

How are you specifying your hyperplane?


oops, accidentally replied to the wrong post.

The hyperplane is specified by a set of orthogonal axes and an origin

Let A be the (D-K) by D matrix of direction cosines
of the hyperplane
axes with the original axes, and let b be the
hyperplane origin in the
original space. Then the projection of a point x in
the original space
onto the hyperplane coordinates is A(x-b).

You want the distribution of Y|Z=0.

isn't projecting the gaussian onto the hyperlane in this way,
equivalent to marginalising the gaussian, which is not really
what I want.

What I want is more like finding the intersection of the gaussian
with the hyperplane, which can be done by conditioning the gaussian
given some of the parameters are known. Provided I can rotate the
gaussian so that points projected onto the hyperplane are treated
as unknown parameters and the rest as known parameters.

You want the distribution of Y|Z=0.

Yes, P(Y|Z=0) is basically what I am looking for, sorry I couldn't
describe it clearly in the first post. I am guessing that the actual
order of the rows in B is not important.

Is there an analytical solution or even robust numerical solution
to this problem?