  We study a non-parametric approach to multivariate density estimation. The
estimators are piecewise constant density functions supported by binary
partitions. The partition of the sample space is learned by maximizing the
likelihood of the corresponding histogram on that partition. We analyze the
convergence rate of the sieve maximum likelihood estimator, and reach a
conclusion that for a relatively rich class of density functions the rate does
not directly depend on the dimension. This suggests that, under certain
conditions, this method is immune to the curse of dimensionality, in the sense
that it is possible to get close to the parametric rate even in high
dimensions. We also apply this method to several special cases, and calculate
the explicit convergence rates respectively.
