In this section we give a detailed proof for the theorem used in Section 3.6.
where
Let U = M-X.
where Y = X-Q. Now, we would like to make the argument of the exponential inside the integral have the following quadratic form so the integration can be done easily:
Equating terms, it is clear that we must have
With these substitutions, we find
The integrand now looks like a Gaussian distribution with mean -A and covariance matrix G; however, the
normalization factor is missing. Therefore, the integral will be 
.
Hence,
Now we will focus on the term 
.
We can use the identity 
to simplify this
equation further.
Therefore,
Resubstituting Y = X- Q establishes the theorem.
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