Frequentist coverage of adaptive nonparametric Bayesian credible sets

Abstract

We investigate the frequentist coverage of Bayesian credible sets in a nonparametric setting. We consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of "polished tail" parameters, in the sense of high probability of coverage of such parameters. On the negative side, we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale.Comment: Published at http://dx.doi.org/10.1214/14-AOS1270 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayes procedures for adaptive inference in inverse problems for the white noise model

“…A combination of elements of the proof of Theorem 5.1 of [22] and new results on the coverage of credible sets from the paper [36] lead us to conjecture that for linear functionals L with coefficients l i i −q−1/2 for some q < p and β > −q there exists a μ 0 ∈ S β such that along a subsequence n j ,…”

“…Since n −(β+q)/(1+2β+2 p) j tends to zero at a slower rate than the minimax rate n −(β+q)/(2β+2 p) for S β , this means that there exist "bad truths" for which the adjusted empirical Bayes procedure does not concentrate at the optimal rate along a subsequence. For linear functionals (2.9) the empirical Bayes posterior Πα n −1/2 (·|Y ) seems only to contract at an optimal rate for "sufficiently nice" truths, for instance of the form μ 0,i i −1/2−β , or the more general polished-tail sequences considered in [36].…”

Fast sampling with Gaussian scale mixture priors in high-dimensional regression

“…Although Bayesian procedures provide an automatic characterization of uncertainty, the resulting credible intervals may not possess the correct frequentist coverage in nonparametric/high-dimensional problems (Szabó et al, 2015). This led us to investigate the frequentist coverage of shrinkage priors in p > n settings; it is trivial to obtain element-wise credible intervals for the β j s from the posterior samples.…”

Objectively combining AR5 instrumental period and paleoclimate climate sensitivity evidence

“…Of course, a Bayesian could use different informative priors at different t, and again, we would have no guarantee of calibration. Priors with good coverage properties are generally viewed as desirable, and determining such priors is still a major topic in Bayesian inference; see for example Szabó et al (2015). If, however, a Bayesian uses (whether intentionally or not) an informative prior, that is not overwhelmed by the data, then the Bayesian posterior density is unlikely to generate a credible interval that is also a confidence interval.…”