  Stochastic partition models divide a multi-dimensional space into a number of
rectangular regions, such that the data within each region exhibit certain
types of homogeneity. Due to the nature of their partition strategy, existing
partition models may create many unnecessary divisions in sparse regions when
trying to describe data in dense regions. To avoid this problem we introduce a
new parsimonious partition model -- the Rectangular Bounding Process (RBP) --
to efficiently partition multi-dimensional spaces, by employing a bounding
strategy to enclose data points within rectangular bounding boxes. Unlike
existing approaches, the RBP possesses several attractive theoretical
properties that make it a powerful nonparametric partition prior on a
hypercube. In particular, the RBP is self-consistent and as such can be
directly extended from a finite hypercube to infinite (unbounded) space. We
apply the RBP to regression trees and relational models as a flexible partition
prior. The experimental results validate the merit of the RBP {in rich yet
parsimonious expressiveness} compared to the state-of-the-art methods.
