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

Abstract

We thank the discussants for their supportive comments and interesting observations. Many questions are still open and not all methodological or philosophical questions may have an answer. Our reply addresses only a subset of questions and is organized by topic. A final section reviews recent work.

Adaptive Bernstein–von Mises theorems in Gaussian white noise

“…The posterior contains significantly more information than merely the 2 or H(δ) norm of the parameter of interest, as can be seen by it assigning mass 1 to a strict subset of 2 . For further discussion on plotting such credible sets see [10,36,49].…”

“…Numerical examples of Cn and C 2 n are displayed in Section 6. Given the similarity of Cn and C 2 n in Figures 2 and 4, a natural question (voiced for example in [10,39,49]) is to what extent these sets actually differ, both in theory and practice. From a purely geometric point of view these sets can be considered as infinite-dimensional ellipsoids with differing orientations.…”

“…The only difference is for the credibility statement, where we no longer have an exponential inequality like Lemma 8.3. However, arguing as in [29] with the H βn -norm instead of the 2 -norm, one can show that under self-similarity the posterior contracts about the posterior mean at rate Mn √ log n for any Mn → ∞ (see also Theorem 1.1 of [49]). It then follows that the second constraint in (4.3) is satisfied with credibility 1 − o P 0 (1).…”