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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?

user70160
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  • If $X,Y, ZY$ are independent multivariate Gaussian random variables, the the joint density of $(X,Y,Z)$ is the product of their densities. The expression $a^TA^{-1}a+b^TB^{-1}b+c^TC^{-1}c$ together with the fact that $\exp(x)\exp(y)\exp(z) = \exp(x+y+z)$ suggests that three multivariate Gaussian densities are being multiplied together. – Dilip Sarwate Apr 25 '15 at 16:40
  • @Dilip Sarwate Thank you. I had made a sign error in the first formula - this has just been corrected. Actually the question refers to the case where two different Gaussian densities (for the same random vector) get multiplied. The result is a third Gaussian density for that vector, multiplied with a scaling factor (referred to in the given sources as normalization constant, although I believe calling it reciproke of the normalization constant would be more intuitive, but I may be wrong). The formulas given above are for the exponent of this scaling factor (-1/2 ignored). – user70160 Apr 25 '15 at 18:00
  • @ Dilip Sarwate FYI - think I found a way - see the answer given, it may not be the shortest or most elegant one, though – user70160 Apr 25 '15 at 23:19

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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).

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