How can I combine the imformation of 2 ore more probablity density functions of a observable which are not gaussian?

Question

We already know that for many gaussian distributions of a observable like N(mu_i,sigma_i) which referring to every measurement we have done, we can use the so-called error-propagation formula like to get a new standard error for the observable considering every measurements before. But if we are not satisfied with just getting such a result, we want a final PDF for this observable, how can we do? Furthermore, if the measurement even does not give a gaussian distribution, how can we do to get a final distribution containing all the information? Thanks.

I actually have no idea now.