Multivariate Stein Factors for a Class of Strongly Log-concave Distributions

2015/12/23 by Lester Mackey, Mackey, Lester, Jackson Gorham +1
Mathematics · #Benford’s Law and Fraud Detection #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #math.PR

paper · pdf · doi:10.48550/arxiv.1512.07392

14 pages. The strong continuity argument in an earlier version did not identify an appropriate Banach space; this version does so. arXiv admin note: substantial text overlap with arXiv:1506.03039

openalex publication_date 2015/12/23 · arxiv created 2016/11/23 · arxiv updated 2016/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions. These "Stein factor" bounds deliver control over Wasserstein and related smooth function distances and are well-suited to analyzing the computable Stein discrepancy measures of Gorham and Mackey. Our arguments of proof are probabilistic and feature the synchronous coupling of multiple overdamped Langevin diffusions.

Citations

• Measuring Sample Quality with Stein's Method

Related

• Stein's density method for multivariate continuous distributions
• Stein operators, kernels and discrepancies for multivariate continuous distributions
• Distances between distributions via Stein's method
• One-dimensional Stein's method with bespoke derivatives
• Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions