Asymptotic Normality of Posterior distribtuion

Question

I am trying to understand asymptotic Normality of posterior distributuions. Specifically, what are the (regularity) conditions that allow us to use this approximation? The problem is that the references I could find online are very mathematical for me to understand clearly. For example the 11th chapter in the book by Hartigan (1983), Bayes Theory; or the paper by Walker (1969).

Please suggest some references to understand asymptotic Normality, for someone without rigorous background in Mathematics. I do have a good understanding of Probability theory.

Answer 1

The asymptotic normality of the posterior distribution IS a technical mathematical result, and the conditions are not necessarily that intuitive. This asymptotic result is known as the "Bernstein–von Mises Theorem". The most "textbook-like" reference I am aware of is:

Vaart, A.W. van der (1998). "10.2 Bernstein–von Mises Theorem". Asymptotic Statistics. Cambridge University Press.

Just give it some time to digest the theory and try to do further readings on the aspects that you do not understand (e.g. convergence, and etcetera).

You can also have a look at the following lecture notes (pp 35):

STATISTICAL THEORY