# Quotient of two Gaussian densities

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.

• did you ever get anywhere with this? Dec 27, 2018 at 14:22

$$\frac{\mathcal{N}(x|\mathbf{m}_1, \Sigma_1)}{\mathcal{N}(x|\mathbf{m}_2, \Sigma_2)} = \mathcal{N}(x|\mathbf{m}, \Sigma) \cdot \mathcal{Z}$$ 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 $$\mathcal{Z} = \frac{|\Sigma_2|}{|\Sigma_2-\Sigma_1|}\cdot \frac{1}{\mathcal{N}(\mathbf{m}_1|\mathbf{m}_2, \Sigma_2-\Sigma_1)}.$$