On the Bernstein-von Mises phenomenon in the Gaussian white noise model

By Haralambie Leahu. 2011

72 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/802974a7b57c514c754f37192fa626c73a4050dc

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Citations in the corpus

BAYESIAN INVERSE PROBLEMS WITH GAUSSIAN PRIORS (2011, unknown): “26 92 v2 [ m at h. ST ] 2 3 Fe b 20 12 The Annals of Statistics 2011, Vol. 39, No. 5, 2626–2657 DOI: 10.1214/11-AOS920 c© Institute of Mathematical Statistics, 2011 BAYESIAN INVERSE PROBLEMS WITH GAUSSIAN PRIORS By B. T. Knapik1, A. W. van der Vaart and J. H. van Zanten”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, background): “It follows from Theorem 2 in [29] that already in the Gaussian conjugate situation, such a result is impossible unless one restricts to very specific priors (which in particular do not possess the plug-in property that will be needed in the key applications below).”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, background): “Finally, although there is no immediate analogue of the BvM property for infinite dimensional parameters, as pointed out by Cox [16] and Freedman [19], some recent contributions have introduced possible notions of nonparametric BvM; see Castillo and Nickl [13] and also Leahu [29].”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “Leahu [20] showed that if the prior variances are chosen very big so that the support of the prior extends beyond 2, then coverage can be obtained.”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, background): “Notably, Freedman [14] has shown that in a basic Gaussian conjugate 2-sequence space setting, the BvM theorem does not hold true in generality; see also the related recent contributions [24, 28].”
Critical dimension in the semiparametric Bernstein—von Mises theorem (2014, background): “Some results for particular semi- and nonparametric problems are available in [18, 17, 22, 8].”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “This dramatically oversmoothes the truth resulting in far too narrow credible bands and is highly dangerous since it is known that oversmoothing the truth can yield zero coverage [27, 29].”
Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, background): “…(BvM) theorem under the total variation metric holds fairly generally and guarantees asymptotically nominal coverage of Bayesian credible intervals, the story of the frequentist validity of credible intervals/bands in inﬁnite dimensional models is more complicated and delicate [16, 20, 24, 27].”