Algebraic structure of Gaussian PDFs

The answer to this question shows that the product of two Gaussian PDFs is also a Gaussian PDF.

My questions are:

• Is there a multiplicative identity for this product?
• More generally what algebraic structure (e.g. semigroup/monoid/ring/field) do Gaussian PDFs belong to?
• OK, `algebraic structure' then. Edited accordingly. – NietzscheanAI Jun 9 '16 at 15:53
• Are you allowed to include delta functions in the structure? This is necessary to have an identity in the structure. – Paul Jun 9 '16 at 15:55
• Uh...the product of two Gaussian PDFs is a Gaussian PDF in a different dimensional space. This seems pretty bad from the perspective of algebra. – Ian Jun 9 '16 at 15:55
• Algebra seems to be able to handle such things with matrices fairly well... – NietzscheanAI Jun 9 '16 at 15:57
• Oh, in that case, then it's commutative, associative, and the Dirac delta is an identity. There are no inverses, and there's no obvious "secondary" operation here to make it into a ring or anything like that. – Ian Jun 9 '16 at 17:24

I will start with the 1-dimensional case. In the question you linked to, they have the identity:

$$N(\mu_1, \sigma_1^2) \times N(\mu_2, \sigma_2^2) \propto N \left( \frac{\sigma_1^2 \mu_2 + \sigma_2^2 \mu_1}{\sigma_1^2 + \sigma_2^2}, \frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}} \right)$$

where the multiplication is of PDFs, not random variables.

Introduce alternate parameters $\lambda_i = 1/\sigma_i^2$ and $\xi_i = \mu_i / \sigma_i^2$. Note that we can recover the original parameters as $\mu_i = \xi_i / \lambda_i, \sigma^2_i = 1/\lambda_i$. This parameterization is called canonical or information form. Note that:

$$N \left( \frac{\sigma_1^2 \mu_2 + \sigma_2^2 \mu_1}{\sigma_1^2 + \sigma_2^2}, \frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}} \right) = N \left( \frac{\xi_1 + \xi_2}{\lambda_1 + \lambda_2}, \frac{1}{\lambda_1 + \lambda_2} \right)$$

Hence the new canonical parameters are $\lambda' = \lambda_1 + \lambda_2, \xi' = \xi_1 + \xi_2$. Since $\lambda \in \mathbb{R}_{>0}$ and $\xi \in \mathbb{R}$, I suppose that in the 1D case you want the Cartesian product of those spaces under addition, $\mathbb{R}_{>0} \times \mathbb{R}$. As $\mathbb{R}_{>0}$ does not support an additive identity, I think this is a semigroup.

In multiple dimensions, canonical form is given by $\bf{\Lambda} = \bf{\Sigma}^{-1}, \bf{\xi} = \bf{\Sigma}^{-1}\bf{\mu}$. I think the addition properties still hold, and you would want to restrict $\bf{\Sigma}$ or equivalently $\bf{\Lambda}$ to be (symmetric) positive definite to avoid degeneracy.

Source: Kevin P. Murphy, Machine Learning, 2nd. ed., section 4.3.3.

Update: I have changed the $=$ in the first equation to $\propto$ because there is a proportionality constant: https://www.johndcook.com/blog/2012/10/29/product-of-normal-pdfs/. Murphy and the linked question both overlook this.

• I posted an answer assuming the operation is multiplication. Gaussian PDFs are clearly not closed under addition, but they are closed under convolution. However, I'm not sure at the moment whether or not there are good distributive laws between the two operations (convolution and multiplication), see mathoverflow.net/questions/3455/… – Brian Z Feb 12 '18 at 18:48