@Journal Article{Summon-FETCH-LOGICAL-c1543-7d60b917a2943e7322d3c5daa86fc58965c836bd0703c8005cade65b4323a3c30,
title = {Adaptive Bayesian Estimation in Indirect Gaussian Sequence Space Models},
author = {Jan Johannes} and {Anna Simoni} and {Rudolf Schenk},
journal = {Annals of economics and statistics},
publisher = {GENES},
year = {2020},
keywords ={Minimax, Economic models, Inverse problems, Sample size, Threshing, Bayes estimators, Random variables, Statistics, Frequentism, Oracles, Economics and Finance, Methods and statistics, Humanities and Social Sciences},
issn ={2115-4430, 1968-3863},
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
that
generates the data. While this establishes posterior consistency, the
concentration rate depends on both
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
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
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
nor on the given
class. Moreover, convergence of the fully data-driven Bayes estimator at the
oracle or at the minimax rate is established.
: C11, C14.
},
}