Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities

• 成果类型: Article
• 署名作者: Cao, Jian; Katzfuss, Matthias
• 署名单位: University of Houston System; University of Houston; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.2025.2546586
• 发表日期: 2026-01-02
• 页码: 768-781
• 关键词: Censored Gaussian process; Integration variable reordering; Minimax exponential tilting; Sparse inverse Cholesky approximation; Truncated multivariate normal; random-fields; prediction

摘要

Multivariate normal (MVN) probabilities arise in myriad applications, but they are analytically intractable and need to be evaluated via Monte Carlo-based numerical integration. For the state-of-the-art minimax exponential tilting (MET) method, we show that the complexity of each of its components can be greatly reduced through an integrand parameterization that uses the sparse inverse Cholesky factor produced by the Vecchia approximation, whose approximation error is often negligible relative to the Monte Carlo error. Based on this idea, we derive algorithms that can estimate MVN probabilities and sample from truncated MVN distributions in linear time (and that are easily parallelizable) at the same convergence or acceptance rate as MET, whose complexity is cubic in the dimension of the MVN probability. We showcase the advantages of our methods relative to existing approaches using several simulated examples. We also analyze a groundwater-contamination dataset with over 20,000 censored measurements to demonstrate the scalability of our method for partially censored Gaussian-process models. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.