HalfStudentT distribution for the Likelyhood. The derivative is zero problem

The topic of model checking can be a bit involved because it is often very dependent on your goals (which I am still a bit confused about here). I would check the Diagnostics and Model Criticism notebooks available here. You can also check out several talks from last year’s PyMCon, like this one and this one. For a deeper dive, you can check out Aki’s keynote.