  In this paper, we study a class of non-parametric density estimators under
Bayesian settings. The estimators are piecewise constant functions on binary
partitions. We analyze the concentration rate of the posterior distribution
under a suitable prior, and demonstrate that the rate does not directly depend
on the dimension of the problem. This paper can be viewed as an extension of a
parallel work where the convergence rate of a related sieve MLE was
established. Compared to the sieve MLE, the main advantage of the Bayesian
method is that it can adapt to the unknown complexity of the true density
function, thus achieving the optimal convergence rate without artificial
conditions on the density.
