Oracle convergence rate of posterior under projection prior and Bayesian model selection

A. Babenko, E.N. Belitser

    Research output: Contribution to journalArticleAcademicpeer-review

    Abstract

    We apply the Bayes approach to the problem of projection estimation of a signal observed in the Gaussian white noise model and we study the rate at which the posterior distribution concentrates about the true signal from the space ℓ2 as the information in observations tends to infinity. A benchmark is the rate of a so-called oracle projection risk, i.e., the smallest risk of an unknown true signal over all projection estimators. Under an appropriate hierarchical prior, we study the performance of the resulting (appropriately adjusted by the empirical Bayes approach) posterior distribution and establish that the posterior concentrates about the true signal with the oracle projection convergence rate. We also construct a Bayes estimator based on the posterior and show that it satisfies an oracle inequality. The results are nonasymptotic and uniform over ℓ2. Another important feature of our approach is that our results on the oracle projection posterior rate are always stronger than any result about posterior convergence with the minimax rate over all nonparametric classes for which the corresponding projection oracle estimator is minimax over this class. We also study implications for the model selection problem, namely, we propose a Bayes model selector and assess its quality in terms of the so-called false selection probability.
    Original languageEnglish
    Pages (from-to)219-245
    Number of pages27
    JournalMathematical Methods of Statistics
    Volume19
    Issue number3
    DOIs
    Publication statusPublished - 2010

    Fingerprint

    Dive into the research topics of 'Oracle convergence rate of posterior under projection prior and Bayesian model selection'. Together they form a unique fingerprint.

    Cite this