https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf contains expressions (p.2, 6e, 6f) for the normalization constant for the product of two multivariate Gaussian pdfs, with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively. The product has mean $c$ and covariance matrix $C$. To simplify the notation, I'm writing here only the exponent for the normalization factor, and ignore the -1/2:
$a^TA^{-1}a+b^TB^{-1}b-c^TC^{-1}c$
Another source, http://www.gaussianprocess.org/gpml/chapters/RWA.pdf, gives the following equation for the same: (page 200, A.8):
$(a-b)^T(A+B)^{-1}(a-b)$
After a number of failed attempts to derive the latter from the former, I think I'm missing a point somewhere.
Is there a straightforward solution? Any hint that could help? Or any source available on the web with the derivation?
Best Answer
One way is to first expand the right summand in the first equation - i.e. perform the vector-matrix multiplications.
Then modify the result using for example the identity given here: Inverse of a sum of symmetric matrices.
Further modify using the matrix inversion theorem (here a special case: $U=V=I$) - it's actually shown on the next page of one of the sources given in the question:http://www.gaussianprocess.org/gpml/chapters/RWA.pdf, p.201 1st paragraph)
What one is left with after further rearranging and simplifying is the second equation plus the two left summands of the first equation - so plugging this result into the first equation and performing the subtractions gives the second equation (times -1, which had been ignored here).