In this section we give a detailed proof for the theorem used in Section 3.6.

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where
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Let U = M-X.
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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:
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Equating terms, it is clear that we must have
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With these substitutions, we find
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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,
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Now we will focus on the term .
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We can use the identity to simplify this
equation further.
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Therefore,
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Resubstituting Y = X- Q establishes the theorem.
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910
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Tue Jan 7 15:44:13 PST 1997