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The Bernstein-van Mises Theorem says that in parametric models under some regularity conditions the posterior mass will contract around the true parameter $\theta_0$ with the optimal frequentist rate independently from the choice of the prior distribution. In nonparametric model case it was shown, that by bad choice of the prior distribution the posterior distribution won't contract at all, or even if it contracts around the true $\theta_0$ the contraction rate will be slower than the optimal frequentist rate. An attempt to solve this problem is to work with a family of prior distributions instead of a single one. It arises the question how to choose the optimal prior distribution out of the family of distributions in the Bayes method. One solution is to put a hyperprior on the family of prior distributions and work with this two level, hierarchical prior distribution. A more practical approach is to choose with an empirical method the optimal prior distribution, this method is called the empirical Bayes method.
We work with the well know white noise model under some regularity assumptions on the unknown, infinite dimensional parameter $\theta_0$. In the Bayes approach we put a family of infinite dimensional Gaussian priors on the parameter set $\Theta$ and we show that we can separate two regions according the smoothness of the parameter $\theta_0$, where in the first one the Empirical Bayes method gives the optimal contraction rate, while in the second one it gives a slower contraction rate than the optimal.
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