If we multiply 2 Gaussian distributions, we will get also a Gaussian distribution, and the precision of a product is the sum of precisions of 2 Gaussians: $\frac{1}{\sigma^2} = \frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}$
(e.g. see here https://ccrma.stanford.edu/~jos/sasp/Product_Two_Gaussian_PDFs.html)
So the resulting precision is higher than any of precisions of any two Gaussians. This seems strange for me if we want to use Gaussians to update our beliefs according to Bayes rule.
For example we have a prior gaussian belief about parameter $\theta$, say with mean $\mu_1 = 0$ and very high precision $\frac{1}{\sigma_1^2} = 100$, also have the likelihood (Gaussian too) with $\mu_2 = 50$ and also high precision $\frac{1}{\sigma_2^2} = 100$. Then the posterior is proportional to the product of prior and likelihood, therefore should have the precision $\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2} = 100 + 100 = 200$. So there is a situation where we are both confident in prior and likelihood, they contradict each other (means are very different). According to my intuition the precision should decrease: when our prior strongly contradicts measurement (likelihood) and we were certain in both, then we should be confused and relax our confidence in posterior, but here we become even more certain in posterior. Could someone point out why my intuition might be wrong?
