I would like to fit multiple distributions that share one of their parameters. As a simple example, let's say I have two different datasets which I know follow a Gaussian distribution. let's say I know $\mu_1$ and $\mu_2$, and now I want to estimate $\sigma_1$ and $\sigma_2$. However, I know that $\sigma_1=\sigma_2$.
Now, I would like to find this joint parameter $\sigma$ that best fits both datasets. The maximum likelihood approach tells us how to find parameters by maximizing the likelihood of one distribution, but how do I maximize the likelihood of both? Perhaps I should maximize the sum of the likelihoods or the product of the likelihoods?