I'm working on a paper about using graphical models to do some prediction tasks with known observations. Since the model is complicated, finding the maximum a posteriori on the true posterior distribution is intractable. Thus, I applied a nested Laplace approximation method to find an approximated MAP with regards to the model parameters. However, the reviewer raised a question and mentioned that I should somehow estimate the distance between the "approximated posterior" and the "true posterior".
I've searched around the distance/similarity metrics comparing two pdfs, but all of them require integral which I have no way to do along with an intractable pdf.
I was thinking about using sampling methods to do an approximation on KL-divergence but not quite clear whether if I can or how to do it. Is there a general way to achieve this task, comparing approximated posterior and true posterior distributions?
Thanks!!