Talk:Bernstein–von Mises theorem

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Mathematics Mid‑priority

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This article seems to be the antithesis of a mathematics article -- actively failing to explain the topic at hand. Can somebody please give a more informative treatment of the topic? — Preceding unsigned comment added by 64.16.138.170 (talk) 20:39, 25 July 2012 (UTC)[reply]

Regarding the quotation, I think it is a polemic that runs against what has been explained before, without further comment. That is an odd choice to conclude this Wikipedia entry. --77.9.44.243 (talk) 12:22, 14 November 2021 (UTC)[reply]

This is currently a page about a theorem, which does not even contain a formal theorem statement... I'm not a Bayesian, so I don't want to dive into fixing this myself, lest I do it wrong (especially regarding the conditions), but in its current form the page is unacceptably bad. 50.93.222.25 (talk) 18:13, 29 January 2022 (UTC)[reply]

Error in theorem consequences

Currently, the introduction of the page says "In particular, it states that Bayesian credible sets of a certain credibility level $\alpha$ will asymptotically be confidence sets of confidence level $\alpha$ , which allows for the interpretation of Bayesian credible sets."

This is trivially false as written. Consider data $X_{1},\ldots ,X_{n}\,{\overset {iid}{\sim }}\,{\mathcal {N}}(\mu _{0},1)$ with prior distribution ${\mathcal {N}}(0,1)$ . We can then consider the set $\mathbb {R} \setminus \{\mu _{0}\}$ , which is obviously a 0% confidence set despite being a 100% credible set for every sample size $n$ . The given statement also appears to me to be in direct contradiction to the False Confidence Theorem, which states that for any bounded, continuous posterior distribution $\pi (\cdot \,|\,\mathbf {X} )$ on the parameter space $\Theta$ , there exists a set $A\subseteq \Theta$ such that $\Pr _{\mathbf {X} \,|\,\theta _{0}}(\pi (A\,|\,\mathbf {X} )\geq 1-\alpha )\geq 1-\alpha$ and yet $\theta _{0}\not \in A$ .

I suspect that this is a result of the statement sounding about right given the informal statement of the theorem presented in the article, while a formal theorem statement would make clear that the above cannot be correct. Aopus (talk) 16:57, 12 September 2024 (UTC)[reply]

I've updated the page to include the formal theorem statement (from Theorem 10.1 of van der Vaart). Indeed, the posterior distribution only converges in total variation distance to the usual asymptotic distribution

{\mathcal {D}}_{n}

of the MLE. Of course, not every set

A

such that

{\mathcal {D}}_{n}(A)=1-\alpha

is an (asymptotic)

(1-\alpha )

-level confidence set, causing the confusion.

I've updated the introduction of the page to say "asymptotically, many Bayesian credible sets... will act as confidence sets" which, while more vague, is technically correct. Additionally, I've given the formal theorem statement and updated the section talking about maximum likelihood estimation.

The "implications" section could still be made more precise, but for now it seems that the page appears to be free of downright falsehoods. Aopus (talk) 21:11, 29 September 2024 (UTC)[reply]