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Mathematical Supplement

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

theorem1406

eqnarray1412

where

eqnarray1432

Let U = M-X.

eqnarray1442

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:

eqnarray1466

Equating terms, it is clear that we must have

eqnarray1472

With these substitutions, we find

eqnarray1482

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 tex2html_wrap_inline3242 . Hence,

eqnarray1503

Now we will focus on the term tex2html_wrap_inline3244 .

eqnarray1517

We can use the identity tex2html_wrap_inline3246 to simplify this equation further.

eqnarray1540

Therefore,

eqnarray1549

Resubstituting Y = X- Q establishes the theorem.

910



Markus Weber
Tue Jan 7 15:44:13 PST 1997