Estimating Density Models with Truncation Boundaries Using Score Matching

Abstract

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 that derive a tractable SM objective are not satisfied by truncated densities. This paper studies 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 domain's boundary. We show this choice of weight function naturally arises from minimizing the Stein discrepancy and upper bounding the finite-sample estimation error. We demonstrate the usefulness of our method via 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.