Estimating Density Models with Truncation Boundaries using Score Matching

Song Liu; Takafumi Kanamori; Daniel J. Williams

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arXiv · 1910.03834

Estimating Density Models with Truncation Boundaries using Score Matching

Song Liu·Takafumi Kanamori·Daniel J. Williams

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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 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.

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BibTeX RIS

Song Liu, Takafumi Kanamori, Daniel J. Williams. 2019-10-09. Estimating Density Models with Truncation Boundaries using Score Matching. https://arxiv.org/abs/1910.03834

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