Accuracy of Gaussian approximation in nonparametric Bernstein -- von Mises Theorem

Understand

The prominent Bernstein -- von Mises (BvM) result claims that the posterior distribution after centering by the efficient estimator and standardizing by the square root of the total Fisher information is nearly standard normal.

• In particular, the prior completely washes out from the asymptotic posterior distribution.
• This fact is fundamental and justifies the Bayes approach from the frequentist viewpoint.
• In the nonparametric setup the situation changes dramatically and the impact of prior becomes essential even for the contraction of the posterior; see [vdV2008], [Bo2011], [CaNi2013,CaNi2014] for different models like Gaussian regression or i.i.d.