- Eingeladener Vortrag (.pdf):
- Mini-Symposium “Statistics on complex structures” der Jahrestagung der Deutschen Mathematiker Vereinigung 2015 in Hamburg
- Vorgetragen von:
- Jan JOHANNES
- Titel:
- Adaptive Bayesian estimation in indirect Gaussian sequence space models
- Abstrakt:
- In an indirect Gaussian sequence space model lower and upper bounds are derived for the concentration rate of the posterior distribution of the parameter of interest shrinking to the parameter value θ◦ that generates the data. While this establishes posterior consistency, however, the concentration rate depends on both θ◦ and a tuning parameter which enters the prior distribution. We first provide an oracle optimal choice of the tuning parameter, i.e., optimized for each θ◦ separately. The optimal choice of the prior distribution allows us to derive an oracle optimal concentration rate of the associated posterior distribution. Moreover, for a given class of parameters and a suitable choice of the tuning parameter, we show that the resulting uniform concentration rate over the given class is optimal in a minimax sense. Finally, we construct a hierarchical prior that is adaptive. This means that, given a parameter θ◦ or a class of parameters, respectively, the posterior distribution contracts at the oracle rate or at the minimax rate over the class. Notably, the hierarchical prior does not depend neither on θ◦ nor on the given class. Moreover, convergence of the fully data-driven Bayes estimator at the oracle or at the minimax rate is established.
- Literatur:
- Johannes, J., Simoni, A. and Schenk, R. (2015). Adaptive Bayesian estimation in indirect Gaussian sequence space models. Discussion paper, arXiv:1502.00184.
| Statistik-Gruppe | Institut für Mathematik | Fakultät für Mathematik und Informatik | Universität Heidelberg |