We study a class of non-parametric density estimators under Bayesian settings. The estimators are obtained by adaptively partitioning the sample space. Under a suitable prior, we analyze the concentration rate of the posterior distribution, and demonstrate that the rate does not directly depend on the dimension of the problem in several special cases. Another advantage of this class of Bayesian density estimators is that it can adapt to the unknown smoothness of the true density function, thus achieving the optimal convergence rate without artificial conditions on the density.

Note: Below, I use [#M] for references in the main paper and [#S] for references in the supplement, since these are indexed differently. Summary: This paper proposes and analyzes a Bayesian approach to nonparametric density estimation. The proposed method is based on approximation by piecewise-constant functions over a binary partitioning of the unit cube, using a prior that decays with the size of the partition. The posterior distribution of the density is shown to concentrate around the true density f_0, at a rate depending on the smoothness r of f_0, a measure in terms of how well f_0 can be approximated by piecewise-constant functions over binary partitionings. Interestingly, the method automatically adapts to unknown r, and r can be related to more standard measures of smoothness, such as Holder continuity, bounded variation, and decay rate of Haar basis coefficients.

We study a class of non-parametric density estimators under Bayesian settings. The estimators are obtained by adaptively partitioning the sample space. Under a suitable prior, we analyze the concentration rate of the posterior distribution, and demonstrate that the rate does not directly depend on the dimension of the problem in several special cases. Another advantage of this class of Bayesian density estimators is that it can adapt to the unknown smoothness of the true density function, thus achieving the optimal convergence rate without artificial conditions on the density. Papers published at the Neural Information Processing Systems Conference.