In this section we give a detailed proof for the theorem used in Section 3.6.
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.
Resubstituting Y = X- Q establishes the theorem.