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I'm searching for good and accessible references (books, courses or papers) on Bayesian asymptotic theory. Possibly put in perspective with frequentist asymptotics. Any suggestions?

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I think that you can start with the book by Gosh Bayesian nonparametrics, which contains a chapter on what is going on for finite dimensional case. Namely, it contains a version of Bernstein-von Mises theorem, posterior consistency and Ibragimov and Hasminsky conditions along with other asymptotic results.

From the general statistics perspective there is a chapter in van der Vaart book Asymptotic statistics devoted to Bayesian procedures.

I think that using these two points of view you can get what you want or at least you will be aware of what exactly you want to find. Note, that both these books are writtent from statstical perspective, and have a small intersection with common machine learning approach to Bayesian statistics.

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