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Last edited on
Oct 18, 2021 by JJ
.
Article:
Annals of Economics and Statistics, 137,83-116
dx.doi.org/10.15609/annaeconstat2009.137.0083

Title:
Adaptive Bayesian estimation in indirect Gaussian sequence space models

Authors:
Jan JOHANNES, Anna Simoni (CREST, CNRS) and Rudolf Schenk

Abstract:
In an indirect Gaussian sequence space model we derive lower and upper bounds for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value θ° that generates the data. While this establishes posterior consistency, the concentration rate depends on both θ° and a tuning parameter which enters the prior distribution. We first provide an oracle optimal choice of the tuning parameter, i.e., optimized for each θ° separately. The optimal choice of the prior distribution allows us to derive an oracle optimal concentration rate of the associated posterior distribution. Moreover, for a given class of parameters and a suitable choice of the tuning parameter, we show that the resulting uniform concentration rate over the given class is optimal in a minimax sense. Finally, we construct a hierarchical prior that is adaptive for mildly ill-posed inverse problems. This means that, given a parameter θ° or a class of parameters, the posterior distribution contracts at the oracle rate or at the minimax rate over the class, respectively. Notably, the hierarchical prior does not depend neither on θ° nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.

Preliminary version:
arXiv:1502.00184

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