The matrix cookbook contains formulas for the product of two multivariate Gaussians, but doesn't appear to contain formulas for the quotient of two Gaussians.
$$
\frac{\mathcal{N}(\mathbf{m}_1, \Sigma_1)}{\mathcal{N}(\mathbf{m}_2, \Sigma_2)} = ~??
$$
Note that I'm not looking for the quotient of two random variables (answered here), I just care about the quotient of two PDFs.
Context: I'm trying to derive factor analysis from Roweis and Ghahramani (1999). They apply Bayes rule:
$$
p(\mathbf{x} \mid \mathbf{y}) = \frac{p(\mathbf{y} \mid \mathbf{x}) p(\mathbf{x})}{p(\mathbf{y})} \\
=\frac{\mathcal{N}(C\mathbf{x}, R)\mathcal{N}(0, I)}{\mathcal{N}(0,CC^T + R)} \\
= \frac{\mathcal{N}((R^{-1} + I)^{-1} (R^{-1} C \mathbf{x}), (R^{-1} + I)^{-1})}{\mathcal{N}(0,CC^T + R)} \\
\vdots \\
??? \\
\vdots \\
= \mathcal{N}(\boldsymbol{\beta} \mathbf{y}, I – \boldsymbol{\beta} C), \quad \boldsymbol{\beta} = C^T (C C^T + R)^{-1}
$$
I used the matrix cookbook for the third equality. R is a covariance matrix without any special assumptions at this point in the paper.
Best Answer
It's a late answer, but it might still be helpful for people who are googling this problem: In the PhD thesis of José Miguel Hernández-Lobato, Appendix C.2 (p. 133) you find the solution to your problem, so you could cite it properly. More easily understandable is this blog post. It's very similar to the solution for the product of two Gaussian densities. I won't dive into the context you gave, but here's the solution for the quotient of two Gaussian densities:
$$\frac{\mathcal{N}(x|\mathbf{m}_1, \Sigma_1)}{\mathcal{N}(x|\mathbf{m}_2, \Sigma_2)} = \mathcal{N}(x|\mathbf{m}, \Sigma) \cdot \mathcal{Z}$$ where $$\Sigma = (\Sigma_1^{-1}-\Sigma_2^{-1})^{-1}$$ and $$\mathbf{m} = \Sigma(\Sigma_1^{-1}\mathbf{m}_1 - \Sigma_2^{-1}\mathbf{m}_2)$$ furthermore $$\mathcal{Z} = \frac{|\Sigma_2|}{|\Sigma_2-\Sigma_1|}\cdot \frac{1}{\mathcal{N}(\mathbf{m}_1|\mathbf{m}_2, \Sigma_2-\Sigma_1)}.$$