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I was reading through Wasserman's All of Statistics and I came across this property in the Bayesian statistics chapter: Theorem

I think I don't really get what is supposed to be the intuition behind it, and the "appropriate regularity conditions" even less so. I would get it if a connexion between the MAP and the MLE existed because the posterior distribution is proportional to the likelihood times the prior, but why is the posterior approximately normal with the MLE as its mean most of the time?

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    $\begingroup$ Maybe you can look up "Bernstein Von Mises theorem", which is well detailed e.g. in Van der Vaart's "Asymptotic statistics". The intuition behind this result can come from an order two development of the log-posterior distribution around the MAP, and by arguing that the MAP get close to the MLE (under suitable conditions) $\endgroup$
    – Pohoua
    Commented Nov 13, 2023 at 21:05
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    $\begingroup$ A related question regarding those "appropriate regularity conditions" is: Under what conditions will a Bayesian posterior fail to converge to a point mass? $\endgroup$
    – Durden
    Commented Jul 14 at 21:42

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