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A Bernstein-von Mises theorem identifies a centered, scaled posterior distribution with the Gaussian law suggested by an efficient likelihood approximation. In an infinite-dimensional model, this conclusion requires control of prior-induced bias; a positive marginal prior density alone does not state enough assumptions.

A concrete calculation shows the issue. In the Gaussian sequence model, let $Y_k={\theta }_{0k}+n^{-1/2}{g}_k$, with ${\theta }_0\in {\ell }^2$, and assign independent priors ${\theta }_k\sim N(0,{\tau }_k^2)$ with every ${\tau }_k^2>0$ and ${\sum }_k{\tau }_k^2<\infty$. Put $a_{nk}=n{\tau }_k^2/(1+n{\tau }_k^2)$. For a nonzero coefficient vector $l\in {\ell }^2$, the linear functional $L(\theta )={\sum }_k{l}_k{\theta }_k$ has the exact posterior lawThe efficient observation statistic $L(Y)$ exists as a Gaussian noise pairing even though $Y$ itself is not in ${\ell }^2$. dominated convergence gives ${\sum }_k{l}_k^2{a}_{nk}\to \Vert l{\Vert }_2^2$. To center at $L(Y)$, however, one also needsThe remaining noise part of the scaled centering error has variance ${\sum }_k{l}_k^2(1-a_{nk}{)}^2\to 0$. Under this displayed bias condition, the conditional law of $\sqrt{n}[L(\theta )-L(Y)]$ converges in total variation distance, in sampling probability, to $N(0,\Vert l{\Vert }_2^2)$. Without bias control, convergence of posterior variances is insufficient. The calculation is specific to conjugate product Gaussian priors; general nonparametric results impose their own prior, truth and functional regularity hypotheses.^[ref]

Ismaël Castillo studied nonparametric Bernstein-von Mises theorems with Richard Nickl. In the Gaussian white noise model, they constructed Bayesian credible sets with asymptotically exact frequentist coverage under stated assumptions. Their analysis also treats functionals, illustrating how an infinite-dimensional unknown function can still yield well-calibrated inference for particular observable quantities.^[ref]

Letso $T(\theta )=a^T({\theta }_1,\dots ,{\theta }_4)$. In the Gaussian sequence model, put $h_k=\sqrt{n}({\theta }_k-Y_k)$. For each relevant coordinate, the posterior density of $h_k$ relative to the standard-normal density $\phi$ isThe log-Lipschitz assumption impliesThese bounds provide Gaussian-integrable domination, while the ratio converges pointwise to one. Dominated convergence, coordinate independence, and the same argument after multiplying by $e^{t{a}^{T}h}$ show that, under the posterior,almost surely. The supplied moment-generating-function criterion therefore gives the finite-functional Bernstein-von Mises theoremwith uniform convergence of distribution functions.

If $z_{1-\alpha }={\Phi }^{-1}(1-\alpha )$, the posterior quantile defining $R_n$ consequently satisfiesin probability. Under $P_{{\theta }_0}^Y$,for every $n$. Quantile convergence and the Slutsky theorem now yield

Richard Nickl coauthored nonparametric Bernstein-von Mises theorems with Ismaël Castillo for the Gaussian white noise model. Their work explains when Bayesian credible sets can also have asymptotically correct frequentist coverage probability. The result connects the limiting posterior geometry with efficient inference on an unknown function, rather than only a fixed finite-dimensional parameter.^[ref]