On importance sampling and independent Metropolis-Hastings with an unbounded weight function

2024-11-14

AI Summary

This paper investigates the approximation error and bias trade-offs between importance sampling (IS) and independent Metropolis–Hastings (IMH) when the importance weight function is unbounded but possesses finite moments. We introduce a particle-based IMH (PIMH) coupling framework to enable unified analysis. Our contributions include: (i) the first tight total variation bounds for IMH/PIMH under unbounded weights; (ii) proof that common-random-number coupling achieves maximal coupling; (iii) rigorous quantitative comparison of finite-time biases for IS and IMH; (iv) sharp convergence rates in terms of both particle number and iteration count; (v) necessary and sufficient conditions for finite moments of unbiased PIMH estimators; and (vi) characterization of the asymptotic efficiency relationship between regular and unbiased IS in the infinite-particle limit. The core innovation lies in establishing a unified coupling-based analytical paradigm, thereby overcoming a fundamental theoretical bottleneck in Monte Carlo analysis under unbounded weights.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Importance sampling and independent Metropolis-Hastings (IMH) are among the fundamental building blocks of Monte Carlo methods. Both require a proposal distribution that globally approximates the target distribution. The Radon-Nikodym derivative of the target distribution relative to the proposal is called the weight function. Under the weak assumption that the weight is unbounded but has a number of finite moments under the proposal distribution, we obtain new results on the approximation error of importance sampling and of the particle independent Metropolis-Hastings algorithm (PIMH), which includes IMH as a special case. For IMH and PIMH, we show that the common random numbers coupling is maximal. Using that coupling we derive bounds on the total variation distance of a PIMH chain to the target distribution. The bounds are sharp with respect to the number of particles and the number of iterations. Our results allow a formal comparison of the finite-time biases of importance sampling and IMH. We further consider bias removal techniques using couplings of PIMH, and provide conditions under which the resulting unbiased estimators have finite moments. We compare the asymptotic efficiency of regular and unbiased importance sampling estimators as the number of particles goes to infinity.

Problem

• Studying approximation error in importance sampling and PIMH with unbounded weights
• Comparing finite-time biases between importance sampling and IMH methods
• Developing unbiased estimators with finite moments for robust mean estimation

Innovation

• Unbounded weight function with finite moments
• Maximal coupling using common random numbers
• Unbiased estimators via coupling techniques