To my understanding, we are allowed to use improper priors to describe our prior knowledge or the lack of the same, but the posterior distribution has to be proper in order for us to make an inference. Are there any guidelines on how one can check if the posterior is proper after having done convergence diagnostic or is it even possible in pymc to sample from improper distribution?
I have searched around for the answer in this forum without luck.
You can use the pm.Flat distribution in PyMC3 for an improper prior, however its use is discouraged. You can use a weakly informing prior or a strongly informing one if you have prior knowledge. A weakly informing prior might be pm.Normal(0, 5) for a standardised predictor.
As for checking if the posterior is proper, I am not sure on the exact methods. At the least you should check 2.5 and 97.5 percentiles and mean are all finite and reasonable. Also check the rhat convergence statistic.
Here is a good guide for choice of priors