## Abstract

We consider empirical Bayesian inference in the many normal means

model in the situation when the high-dimensional mean vector is multilevel sparse, that is,most of the entries of the parameter vector are some fixed values. For instance, the traditional sparse signal is a particular case (with one level) of multilevel sparse sequences.We apply an empirical Bayesian approach, namely we put an appropriate prior modeling the multilevel sparsity and make data-dependent choices of certain parameters of the prior. We establish local (i.e., with rate depending on the “true” parameter) posterior contraction and estimation results. Global adaptive minimax results (for the estimation and posterior contraction problems) over sparsity classesfollow from our local results if the sparsity level is of polynomial order. The results are illustrated by simulations.

model in the situation when the high-dimensional mean vector is multilevel sparse, that is,most of the entries of the parameter vector are some fixed values. For instance, the traditional sparse signal is a particular case (with one level) of multilevel sparse sequences.We apply an empirical Bayesian approach, namely we put an appropriate prior modeling the multilevel sparsity and make data-dependent choices of certain parameters of the prior. We establish local (i.e., with rate depending on the “true” parameter) posterior contraction and estimation results. Global adaptive minimax results (for the estimation and posterior contraction problems) over sparsity classesfollow from our local results if the sparsity level is of polynomial order. The results are illustrated by simulations.

Original language | English |
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Title of host publication | Bayesian Statistics in Action |

Publisher | Springer |

Pages | 51-66 |

Number of pages | 16 |

Volume | 194 |

ISBN (Electronic) | 978-3-319-54084-9 |

ISBN (Print) | 978-3-319-54083-2 |

DOIs | |

Publication status | Published - 2017 |

### Publication series

Name | Springer Proceedings in Mathematics & Statistics |
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Publisher | Springer |

## Keywords

- Empirical Bayesian approach
- Local posterior concentration rate
- Multilevel sparse sequences