TY - JOUR
T1 - Honest Bayesian confidence sets for the L2-norm
AU - Szabó, Botond
AU - van der Vaart, Aad
AU - van Zanten, Harry
PY - 2015/11/1
Y1 - 2015/11/1
N2 - We investigate the problem of constructing Bayesian credible sets that are honest and adaptive for the L2-loss over a scale of Sobolev classes with regularity ranging between [D, 2D], for some given D in the context of the signal-in-white-noise model. We consider a scale of prior distributions indexed by a regularity hyper-parameter and choose the hyper-parameter both by marginal likelihood empirical Bayes and by hierarchical Bayes method, respectively. Next we consider a ball centred around the corresponding posterior mean with prescribed posterior probability. We show by theory and examples that both the empirical Bayes and the hierarchical Bayes credible sets give misleading, overconfident uncertainty quantification for certain oddly behaving truth. Then we construct a new empirical Bayes method based on risk estimation, which provides the correct uncertainty quantification and optimal size.
AB - We investigate the problem of constructing Bayesian credible sets that are honest and adaptive for the L2-loss over a scale of Sobolev classes with regularity ranging between [D, 2D], for some given D in the context of the signal-in-white-noise model. We consider a scale of prior distributions indexed by a regularity hyper-parameter and choose the hyper-parameter both by marginal likelihood empirical Bayes and by hierarchical Bayes method, respectively. Next we consider a ball centred around the corresponding posterior mean with prescribed posterior probability. We show by theory and examples that both the empirical Bayes and the hierarchical Bayes credible sets give misleading, overconfident uncertainty quantification for certain oddly behaving truth. Then we construct a new empirical Bayes method based on risk estimation, which provides the correct uncertainty quantification and optimal size.
KW - Coverage
KW - Credible sets
KW - Uncertainty quantification
UR - http://www.scopus.com/inward/record.url?scp=84933568599&partnerID=8YFLogxK
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U2 - 10.1016/j.jspi.2014.06.005
DO - 10.1016/j.jspi.2014.06.005
M3 - Article
AN - SCOPUS:84933568599
VL - 166
SP - 36
EP - 51
JO - Journal of Statistical Planning and Inference
JF - Journal of Statistical Planning and Inference
SN - 0378-3758
ER -