Talk (.pdf): : 9th International Conference of the ERCIM Working Group on Computational and Methodological Statistics in Sevilla, Spain

Presented by:

Xavier LOIZEAU

Title:

A Bayesian interpretation of data-driven estimation by model selection

Abstract:

Considering an indirect Gaussian sequence space model and a hierarchical prior, oracle/minimax-optimal concentration and convergence of the associated posterior and Bayes estimator are shown. Notably, the hierarchical prior does not depend neither on the true parameter value nor on the given class. The posterior is taken iteratively as a new prior and the associated posterior is calculated for the same observation again and again. Thereby, a family, indexed by the iteration parameter, of fully data driven priors is constructed. Each element of this family leads to an oracle/minimax-optimal concentration and convergence of the associated posterior and Bayes estimator as the noise level tends to zero. For a fixed noise level letting the iteration parameter tend to infinity, the associated posterior distribution shrinks to a point measure. The limit distribution is degenerated on the value of a projection estimator with fully-daten driven choice of the dimension parameter using a model selection approach where a penalized contrast criterium is minimised. Thereby, the classical model selection approach gives in some sense an infinitely increasing weight to the information contained in the observations in comparison to the prior distribution. It is further shown that the limit distribution and the associated Bayes estimator converges with oracle and minimax-optimal rates as the noise level tends to zero.

References:

Johannes, J., Simoni, A. and Schenk, R. (2015). Adaptive Bayesian estimation in indirect Gaussian sequence space models. Discussion paper, arXiv:1502.00184.

Johannes, J., and Loizeau, X. (2016). A Bayesian interpretation of data-driven estimation by model selection. Discussion paper in preparation.