= Bernstein-von Mises theorem
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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> $L(\theta)=\sum_kl_k\theta_k$ has the exact posterior law
$$
L(\theta)\mid Y\sim N\left(\sum_kl_ka_{nk}Y_k,\frac1n\sum_kl_k^2a_{nk}\right).
$$
The efficient observation statistic $L(Y)$ exists as a Gaussian noise pairing even though $Y$ itself is not in $\ell^2$. <dominated convergence> gives $\sum_kl_k^2a_{nk}\to\|l\|_2^2$. To center at $L(Y)$, however, one also needs
$$
\sqrt n\sum_kl_k(1-a_{nk})\theta_{0k}\longrightarrow0.
$$
The remaining noise part of the scaled centering error has <variance> $\sum_kl_k^2(1-a_{nk})^2\to0$. Under this displayed bias condition, the conditional law of $\sqrt n[L(\theta)-L(Y)]$ converges in <total variation distance>, in sampling <probability>, to $N(0,\|l\|_2^2)$. 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.<Nonparametric Bernstein-von Mises theorems in Gaussian white noise by Ismaël Castillo and Richard Nickl>[2013, Section 2.3.1, linear-function inference; the conjugate-coordinate calculation is derived here]{ref}
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