Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models

Jan Johannes, Anna Simoni and Rudolf Schenk

Annals of Economics and Statistics, 2020, issue 137, 83-116

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 THETA° that generates the data. While this establishes posterior consistency, the concentration rate depends on both THETA° 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 THETA° 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 THETA° 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 THETA° nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.

Keywords: Bayesian Nonparametrics; Sieve Prior; Hierarchical Bayes; Exact Concentration Rates; Oracle Optimality; Minimax Theory; Adaptation. (search for similar items in EconPapers)
JEL-codes: C11 C14 (search for similar items in EconPapers)
Date: 2020
References: Add references at CitEc
Citations: View citations in EconPapers (1)

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Related works:
Working Paper: Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models (2020)
Working Paper: Adaptive Bayesian estimation in indirect Gaussian sequence space models (2015)
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Persistent link: https://EconPapers.repec.org/RePEc:adr:anecst:y:2020:i:137:p:83-116

DOI: 10.15609/annaeconstat2009.137.0083

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