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

Gianluca Finocchio*, Johannes Schmidt-Hieber*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

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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.
Original languageEnglish
Pages (from-to)239-271
Number of pages33
JournalElectronic Journal of Statistics
Issue number1
Publication statusPublished - 8 Jan 2020


  • Bernstein-von Mises theorems
  • Frequentist Bayes
  • Gaussian sequence model
  • Maximum likelihood
  • Semiparametric inference


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