Hello, I am very new to Pymc so I apologise for the very basic question, but this has been a general long term doubt I have about the Marginal implementation of Gaussian Processes. As I understand, we assume a Gaussian likelihood, so we can calculate the marginal likelihood performing the integral directly. Why do we still need to do sampling to get the posterior distribution, couldn’t we just use Bayes Rule to get the posterior distribution given that we have already calculated the integral in the denominator analytically ?

I understand that we use sampling in general to approximate the posterior not the marginal likelihood, but if we have already its ingredients why do we still need to do sampling ? Please re guide me if I am missing something and thanks very much!