I am studying now GLMMs (Generalized Linear Mixed Models). From my understanding, in order to estimate the parameters of this model, you need to arrive at the marginal probability by integrating a conditional one - which often gives intractable integrals without closed form solutions. So approximation methods are used, such as MC (package glmm), or GHQ or Laplace (package glmmML), etc.
But this implies that the marginal model is no longer in the Exponential family.
My question is - does the limitation of the distribution to an exponential family (which exists for regular GLM's) is still required (in GLMM's)? I see that in the R packages you still need to specify a family, but this could be for simplicity and being-used-to reasons. You could just as easily supply the (custom) pdf function to it.
Also, if numerical methods like GLMM exists, what do we actually gain from "vanilla" GLM's? So ok, there's 1 extra layer of approximation saved for having an explicit pdf instead of an integral. Anything else?
Closely related question - here.