# Estimating Density Models with Truncation Boundaries using Score Matching

arXiv: https://arxiv.org/abs/1910.03834 · HTML: https://mlsanity.com/paper/1910.03834/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2019.

## Built on by (most cited first, 20 of 20)

- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Robust and Scalable Bayesian Online Changepoint Detection](https://mlsanity.com/paper/2302.04759/) (2023)
- [Robust and Conjugate Gaussian Process Regression](https://mlsanity.com/paper/2311.00463/) (2023)
- [Score Matching With Missing Data](https://mlsanity.com/paper/2506.00557/) (2025)
- [Efficient Statistics With Unknown Truncation, Polynomial Time Algorithms, Beyond Gaussians](https://mlsanity.com/paper/2410.01656/) (2024)
- [Is Score Matching Suitable for Estimating Point Processes?](https://mlsanity.com/paper/2410.04037/) (2024)
- [Score Matching for Truncated Density Estimation on a Manifold](https://mlsanity.com/paper/2206.14668/) (2022)
- [On Computationally Efficient Learning of Exponential Family Distributions](https://mlsanity.com/paper/2309.06413/) (2023)
- [A Unified View on Learning Unnormalized Distributions via Noise-Contrastive Estimation](https://mlsanity.com/paper/2409.18209/) (2024)
- [High-Dimensional Differential Parameter Inference in Exponential Family using Time Score Matching](https://mlsanity.com/paper/2410.10637/) (2024)
- [Learned Bayesian Cram'er-Rao Bound for Unknown Measurement Models Using Score Neural Networks](https://mlsanity.com/paper/2502.00724/) (2025)
- [Dale meets Langevin: A Multiplicative Denoising Diffusion Model](https://mlsanity.com/paper/2510.02730/) (2025)
- [Multi-Output Robust and Conjugate Gaussian Processes](https://mlsanity.com/paper/2510.26401/) (2025)
- [Approximate Stein Classes for Truncated Density Estimation](https://mlsanity.com/paper/2306.00602/) (2023)
- [Score-Based Change-Point Detection and Region Localization for Spatio-Temporal Point Processes](https://mlsanity.com/paper/2602.04798/) (2026)
- [Interaction Models and Generalized Score Matching for Compositional Data](https://mlsanity.com/paper/2109.04671/) (2021)
- [Event Generation with Parallel Langevin Sampling and Learned Stein Diagnostics](https://mlsanity.com/paper/2606.14854/) (2026)
- [Should the Boundary Term Be Learned in Reflected Diffusion? Conormal Trace and Reflection Masking](https://mlsanity.com/paper/2608.03469/) (2026)
- [Robust Bayesian Inference for Unnormalized Models with Mixed-Domain Data](https://mlsanity.com/paper/2609.01783/) (2026)

## Abstract (arXiv)

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.