I cam across an interesting claim that:

$\mathcal N (\mu_f, \sigma_f^2) \; \mathcal N (\mu_g, \sigma_g^2) = \mathcal N \left(\frac{\mu_f\sigma_g^2+\mu_g\sigma_f^2}{\sigma_f^2+\sigma_g^2}, \frac {\sigma_f^2\sigma_g^2}{\sigma_f^2+\sigma_g^2}\right)$

In trying to understand it I consulted Bromiley:

http://www.tina-vision.net/docs/memos/2003-003.pdf

Bromiley concludes that:

if

$f(x) = \frac{1}{\sqrt{2\pi\sigma_f^2}} e^{-\frac{(x-\mu_f)^2}{2 \sigma_f^2}}$ and $g(x) = \frac{1}{\sqrt{2\pi\sigma_g^2}} e^{-\frac{(x-\mu_g)^2}{2 \sigma_g^2}}$

then:

$f(x)g(x) = D_{fg} \frac{1}{\sqrt{2\pi\sigma_{fg}^2}} e^{- \frac { (x - \mu_{fg})^2 } {2 \sigma_{fg}^2 } }$

where:

$\mu_{fg} = \frac { \sigma_g^2\mu_f + \sigma_f^2 \mu_g } {\sigma_f^2 + \sigma_g^2}$ and $\sigma_{fg}^2 = \frac {\sigma_f^2 \sigma_g^2} {\sigma_f^2 + \sigma_g^2}$

$S_{fg} = \frac {1} {\sqrt{2\pi(\sigma_f^2+\sigma_g^2)}} e^{ -\frac{(\mu_f-\mu_g)^2}{2(\sigma_f^2+\sigma_g^2)} }$

Note that if $\mu_f$, $\mu_g$ , $\sigma_f$ and $\sigma_f$ are known constants then the $S_{fg}$ is a known constant too.

To wit, if I cast Bromiley's result in the format of the claim I'm exploring:

$\mathcal N (\mu_f, \sigma_f^2) \; \mathcal N (\mu_g, \sigma_g^2) = S_{fg} \; \mathcal N \left(\frac{\mu_f\sigma_g^2+\mu_g\sigma_f^2}{\sigma_f^2+\sigma_g^2}, \frac {\sigma_f^2\sigma_g^2}{\sigma_f^2+\sigma_g^2}\right)$

In short there is a constant scaling factor $S_{fg}$. In fact Bromiley describes the product as a scaled Gaussian.

Given $f(x)$ and $g(x)$ are both functions of $x$ the original claim, which reads (as a reminder):

$\mathcal N (\mu_f, \sigma_f^2) \; \mathcal N (\mu_g, \sigma_g^2) = \mathcal N \left(\frac{\mu_f\sigma_g^2+\mu_g\sigma_f^2}{\sigma_f^2+\sigma_g^2}, \frac {\sigma_f^2\sigma_g^2}{\sigma_f^2+\sigma_g^2}\right)$

implies that:

$\int_{-\infty}^{\infty} f(x) g(x) \;dx = 1$

But Bromiley's result suggests this implication is false. I presume it inetgrates to S_{fg}, or:

$\int_{-\infty}^{\infty} f(x) g(x) \;dx = S_{fg}$

My tentative conclusion is that the claim I am exploring is false, and my questions would be:

- Is my tentative conclusion true? (is the explored claim false?)
- Am I right in concluding the integral would be $S_{fg}$?

Those are the areas I'm a little shakey on at present and seek some review on I guess.