  Truncated densities are probability density functions defined on truncated
domains. They share the same parametric form with their non-truncated
counterparts up to a normalizing constant. Since the computation of their
normalizing constants is usually infeasible, Maximum Likelihood Estimation
cannot be easily applied to estimate truncated density models. Score Matching
(SM) is a powerful tool for fitting parameters using only unnormalized models.
However, it cannot be directly applied here as boundary conditions used to
derive a tractable SM objective are not satisfied by truncated densities. In
this paper, we study parameter estimation for truncated probability densities
using SM. The estimator minimizes a weighted Fisher divergence. The weight
function is simply the shortest distance from a data point to the boundary of
the domain. We show this choice of weight function naturally arises from
minimizing the Stein discrepancy as well as upperbounding the finite-sample
estimation error. The usefulness of our method is demonstrated by numerical
experiments and a study on the Chicago crime data set. We also show that the
proposed density estimation can correct the outlier-trimming bias caused by
aggressive outlier detection methods.
