Reviews: Convergence rates of a partition based Bayesian multivariate density estimation method

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.