### Abstract

^{2}is biased and inconsistent. This raises the question whether the posterior is able to correct the MLE in this case. By developing a new proving strategy that uses refined properties of the posterior distribution, we find that the marginal posterior is inconsistent for any i.i.d. prior on the mean parameters. In particular, no assumption on the decay of the prior needs to be imposed. Surprisingly, we also find that consistency can be retained for a hierarchical prior based on Gaussian mixtures. In this case we also establish a limiting shape result and determine the limit distribution. In contrast to the classical Bernstein-von Mises theorem, the limit is non-Gaussian. We show that the Bayesian analysis leads to new statistical estimators outperforming the correctly calibrated MLE in a numerical simulation study.

Original language | English |
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Pages (from-to) | 239-271 |

Journal | Electronic Journal of Statistics |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - 8 Jan 2020 |

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*Electronic Journal of Statistics*,

*14*(1), 239-271. https://doi.org/10.1214/19-EJS1671

}

*Electronic Journal of Statistics*, vol. 14, no. 1, pp. 239-271. https://doi.org/10.1214/19-EJS1671

**Bayesian variance estimation in the Gaussian sequence model with partial information on the means.** / Finocchio, Gianluca ; Schmidt-Hieber, Anselm Johannes.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Bayesian variance estimation in the Gaussian sequence model with partial information on the means

AU - Finocchio, Gianluca

AU - Schmidt-Hieber, Anselm Johannes

PY - 2020/1/8

Y1 - 2020/1/8

N2 - Consider the Gaussian sequence model under the additional assumption that a fixed fraction of the means is known. We study the problem of variance estimation from a frequentist Bayesian perspective. The maximum likelihood estimator (MLE) for σ2 is biased and inconsistent. This raises the question whether the posterior is able to correct the MLE in this case. By developing a new proving strategy that uses refined properties of the posterior distribution, we find that the marginal posterior is inconsistent for any i.i.d. prior on the mean parameters. In particular, no assumption on the decay of the prior needs to be imposed. Surprisingly, we also find that consistency can be retained for a hierarchical prior based on Gaussian mixtures. In this case we also establish a limiting shape result and determine the limit distribution. In contrast to the classical Bernstein-von Mises theorem, the limit is non-Gaussian. We show that the Bayesian analysis leads to new statistical estimators outperforming the correctly calibrated MLE in a numerical simulation study.

AB - Consider the Gaussian sequence model under the additional assumption that a fixed fraction of the means is known. We study the problem of variance estimation from a frequentist Bayesian perspective. The maximum likelihood estimator (MLE) for σ2 is biased and inconsistent. This raises the question whether the posterior is able to correct the MLE in this case. By developing a new proving strategy that uses refined properties of the posterior distribution, we find that the marginal posterior is inconsistent for any i.i.d. prior on the mean parameters. In particular, no assumption on the decay of the prior needs to be imposed. Surprisingly, we also find that consistency can be retained for a hierarchical prior based on Gaussian mixtures. In this case we also establish a limiting shape result and determine the limit distribution. In contrast to the classical Bernstein-von Mises theorem, the limit is non-Gaussian. We show that the Bayesian analysis leads to new statistical estimators outperforming the correctly calibrated MLE in a numerical simulation study.

U2 - 10.1214/19-EJS1671

DO - 10.1214/19-EJS1671

M3 - Article

VL - 14

SP - 239

EP - 271

JO - Electronic Journal of Statistics

JF - Electronic Journal of Statistics

SN - 1935-7524

IS - 1

ER -