Abstract
We study the Bernstein-von Mises (BvM) phenomenon, i.e., Bayesian credible sets and frequentist confidence regions for the estimation error coincide asymptotically, for the infinite-dimensional Gaussian white noise model governed by Gaussian prior with diagonal-covariance structure. While in parametric statistics this fact is a consequence of (a particular form of) the BvM Theorem, in the nonparametric setup, however, the BvM Theorem is known to fail even in some, apparently, elementary cases. In the present paper we show that BvM-like statements hold for this model, provided that the parameter space is suitably embedded into the support of the prior. The overall conclusion is that, unlike in the parametric setup, positive results regarding frequentist probability coverage of credible sets can only be obtained if the prior assigns null mass to the parameter space.
Highlights
In parametric statistics, the celebrated Bernstein-von Mises (BvM) Theorem states that in a statistical model with finite-dimensional parameter θ ∈ Θ, if the observed variable X follows some known distribution Pθ := L(X|θ) and π is a prior probability on the parameter space Θ under fairly general conditions over the true parameter, the model and the prior, the centered Bayesian posterior and the sampling distribution of any asymptotically efficient estimator centered at truth will be close for a large number of observations, where “close” means with respect to total variation norm; see, e.g., [13]
We study the Bernstein-von Mises (BvM) phenomenon, i.e., Bayesian credible sets and frequentist confidence regions for the estimation error coincide asymptotically, for the infinite-dimensional Gaussian white noise model governed by Gaussian prior with diagonal-covariance structure
The celebrated Bernstein-von Mises (BvM) Theorem states that in a statistical model with finite-dimensional parameter θ ∈ Θ, if the observed variable X follows some known distribution Pθ := L(X|θ) and π is a prior probability on the parameter space Θ under fairly general conditions over the true parameter, the model and the prior, the centered Bayesian posterior and the sampling distribution of any asymptotically efficient estimator centered at truth will be close for a large number of observations, where “close” means with respect to total variation norm; see, e.g., [13]
Summary
The celebrated Bernstein-von Mises (BvM) Theorem states that in a statistical model with finite-dimensional parameter θ ∈ Θ, if the observed variable X follows some known distribution Pθ := L(X|θ) and π is a prior probability on the parameter space Θ under fairly general conditions over the true parameter, the model and the prior, the centered Bayesian posterior and the sampling distribution of any asymptotically efficient estimator centered at truth will be close for a large number of observations, where “close” means with respect to total variation norm; see, e.g., [13].
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