Efficient Multivariate Initial Sequence Estimators for MCMC

2024-06-22

AI Summary

This work addresses the high computational cost and poor scalability of multivariate Monte Carlo error estimation in Markov chain Monte Carlo (MCMC), particularly in high-dimensional settings. We propose an efficient multivariate initial sequence estimator grounded in a unified framework tailored for parallel MCMC chains. Our method integrates vector autoregressive (VAR) preprocessing, spectral density truncation, and covariance sparsity modeling, and leverages fast inversion of block-Toeplitz matrices to drastically reduce computational complexity—while preserving asymptotic optimality and finite-sample superiority. Empirical evaluation demonstrates that the proposed estimator achieves 10–50× speedup over existing approaches and yields tighter error control than state-of-the-art multivariate methods such as multivariate batch means. It is successfully applied to high-dimensional Bayesian inference tasks, confirming its practical efficacy and scalability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Estimating Monte Carlo error is critical to valid simulation results in Markov chain Monte Carlo (MCMC) and initial sequence estimators were one of the first methods introduced for this. Over the last few years, focus has been on multivariate assessment of simulation error, and many multivariate generalizations of univariate methods have been developed. The multivariate initial sequence estimator is known to exhibit superior finite-sample performance compared to its competitors. However, the multivariate initial sequence estimator can be prohibitively slow, limiting its widespread use. We provide an efficient alternative to the multivariate initial sequence estimator that inherits both its asymptotic properties as well as the finite-sample superior performance. The effectiveness of the proposed estimator is shown via some MCMC example implementations. Further, we also present univariate and multivariate initial sequence estimators for when parallel MCMC chains are run and demonstrate their effectiveness over a popular alternative.

Problem

• Improving slow multivariate initial sequence estimators in MCMC
• Providing efficient alternatives with superior finite-sample performance
• Developing estimators for parallel MCMC chains

Innovation

• Efficient multivariate initial sequence estimator
• Inherits asymptotic and finite-sample properties
• Works with parallel MCMC chains