- Thesis:
- Bachelor in Mathematics
- Author:
- Julia Renner
- Title:
- Bayesian nonparametric log-concave density estimation
- Supervisor:
- Jan JOHANNES
- Abstract:
- This thesis focuses on log-concave density estimation on R and introduces a nonparametric approach to this problem. We therefore consider observations that are distributed according to an unknown log-concave density which shall be estimated. First, we show a rate of convergence of the posterior distribution based on the maximum likelihood estimator. Second, we introduce a prior based on an exponentiated Dirichlet process mixture. Using the stick-breaking construction as an approach for the Dirichlet process, we illustrate some typical draws. We show a rate of convergence of the posterior distribution for the introduced prior in the case, in which the “true” density has known compact support, as well as in the case, in which the support of the “true” density is unknown. Last, we present a hierarchical construction for the prior and show a rate of convergence of the posterior distribution based on this construction.
References:- E. Mariucci, K. Ray, and B. Szabó. A Bayesian nonparametric approach to log-concave density estimation, Bernoulli, 26(2):1070-1097, 2020.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |