On Bayesian Based Adaptive Confidence Sets for Linear Functionals

Abstract

We consider the problem of constructing Bayesian based confidence sets for linear functionals in the inverse Gaussian white noise model. We work with a scale of Gaussian priors indexed by a regularity hyper-parameter and apply the data-driven (slightly modified) marginal likelihood empirical Bayes method for the choice of this hyper-parameter. We show by theory and simulations that the credible sets constructed by this method have sub-optimal behaviour in general. However, by assuming "self-similarity" the credible sets have rate-adaptive size and optimal coverage. As an application of these results we construct L ∞ -credible bands for the true functional parameter with adaptive size and optimal coverage under self-similarity constraint.

Gaussian approximation of general non-parametric posterior distributions

“…The existing literature mostly focus on functionals where efficient estimation with √ n-rate is available ( [41,7,8,9]). The more general inefficient estimation with slower than root-n rate (e.g., evaluation functional) is only treated recently by [49] in Gaussian white noise model. As will be seen, our theory treat efficient and inefficient estimation in a unified framework.…”

Rejoinder to discussions of “Frequentist coverage of adaptive nonparametric Bayesian credible sets”

“…We have encountered similar phenomena when deriving contraction rates and credible intervals for (not necessarily continuous) linear functionals of the parameter; see [13]. Since point evaluations are linear functionals, such credible intervals can be glued together into L ∞ -credible bands, where due to the Gaussianity one would expect at most a logarithmic factor to be necessary to pass from pointwise to simultaneous intervals.…”

“…The likelihood-based empirical Bayes method seems to "estimate" the Sobolev regularity of the truth. In [13] we have shown that coverage can be retained by subtracting 1/2 from the estimate, thus under-smoothing the empirical Bayes posterior distribution. In forthcoming work with Sniekers, we note that the ordinary empirical Bayes procedure may still give good coverage for many true parameters, the loss of 1/2 being really a worst case comparison of the two norms and coverage being connected to more subtle properties of the true parameter.…”