Gaussian Invariant Markov Chain Monte Carlo

2026-07-14 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, extended

Summary

Gaussian Invariant Markov Chain Monte Carlo (GI-MCMC) methods, including GI-RWM and GI-MALA, are introduced to enhance the statistical efficiency of MCMC sampling. These methods leverage a unique Gaussian invariance property, enabling exact analytical solutions to the Poisson equation for Gaussian targets. These solutions are then used to construct efficient control variates for significant variance reduction in estimators, even for intractable non-Gaussian targets. Experiments demonstrate GI-MALA's superior performance, achieving 1.5 to 3 times higher Effective Sample Size (ESS) than standard MALA in Bayesian logistic regression and the highest Min ESS/s in high-dimensional latent Gaussian models (up to d=4096), achieving leading performance. Optimal acceptance rates for GI-MALA are found to be higher, typically between 75% and 85%, contrasting with MALA's 57.4%. Theoretical analysis confirms geometric ergodicity and provides an optimal scaling analysis that links acceptance rates to target Gaussianity.

Key takeaway

For Machine Learning Engineers and Research Scientists performing high-dimensional Bayesian inference, you should adopt Gaussian Invariant MCMC (GI-MCMC) methods like GI-MALA. These samplers provide significantly improved statistical efficiency and variance reduction by leveraging analytical solutions to the Poisson equation. Tune your GI-MALA step size to achieve higher acceptance rates, typically 75-85%, rather than the standard 57.4% for MALA, to maximize estimator accuracy and accelerate convergence in complex models like latent Gaussian models.

Key insights

Gaussian invariance in MCMC enables exact Poisson equation solutions, significantly reducing estimator variance for both Gaussian and non-Gaussian targets.

Principles

• Gaussian invariance allows exact analytical solutions to the Poisson equation for Gaussian targets.
• Control variates derived from these solutions effectively reduce variance for intractable non-Gaussian targets.
• Optimal acceptance rates for Gaussian invariant MCMC are target-dependent, often exceeding standard MALA rates.

Method

Develop Gaussian invariant proposals (GI-RWM, GI-MALA) leveraging target derivatives. Derive exact or approximate Poisson equation solutions for Gaussian targets. Construct control variates from these solutions for variance reduction in non-Gaussian targets, optimizing parameters via asymptotic variance minimization.

In practice

• Implement GI-MALA with an O(d^2) approximate inverse Hessian preconditioning for high-dimensional latent Gaussian models.
• Utilize truncated Poisson equation solutions to reduce variance when estimating expectations of complex functions like indicator functions.

Topics

• Markov Chain Monte Carlo
• Gaussian Invariant MCMC
• Variance Reduction
• Control Variates
• Latent Gaussian Models
• Optimal Scaling

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.