Quasi-Monte Carlo with one categorical variable

2025/06/19 by Ho, Valerie N. P., Owen, Art B., Pan, Zexin
#Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2506.16582

Abstract

We study randomized quasi-Monte Carlo (RQMC) estimation of a multivariate integral where one of the variables takes only a finite number of values. This problem arises when the variable of integration is drawn from a mixture distribution as is common in importance sampling and also arises in some recent work on transport maps. We find that when integration error decreases at an RQMC rate that it is then beneficial to oversample the smallest mixture components instead of using a proportional allocation. The optimal allocations depend on the possibly unknown convergence rates. Designing the sample with an incorrect assumption on the rate still attains that convergence rate, with an inferior implied constant. The penalty for using a pessimistic rate is typically higher than for using an optimistic one. We also find that for the most accurate RQMC sampling methods, it is advantageous to arrange that our n=2^m randomized Sobol' points split into subsample sizes that are also powers of 2.

Citations

• Integrability of weak mixed first-order derivatives and convergence rates of scrambled digital nets
• Transporting Higher-Order Quadrature Rules: Quasi-Monte Carlo Points and Sparse Grids for Mixture Distributions
• Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation
• Where are the logs?
• On dropping the first Sobol' point
• A strong law of large numbers for scrambled net integration
• A constraint on extensible quadrature rules
• A review on global sensitivity analysis methods
• On Koksma-Hlawka inequality
• The On-Line Encyclopedia of Integer Sequences

Related

• Quasi Importance Sampling
• Quasi-Monte Carlo Methods: What, Why, and How?
• Quasi-Monte Carlo integration over ℝ^s with boundary-damping importance sampling
• Quasi Markov Chain Monte Carlo Methods
• On the error rate of importance sampling with randomized quasi-Monte Carlo