Transformed Samplers with Variance Reduction

2026-10-07

AI Summary

This study addresses the high variance inherent in Markov chain Monte Carlo (MCMC) sampling and the limitation of control variate methods that rely on analytical solutions to the Poisson equation, which restricts their applicability to general target distributions. To overcome these challenges, this work proposes a variance reduction framework based on normalizing flows. By leveraging bijective transformations to map the target distribution into a reference latent space, the authors derive the transformed Markov kernel and an explicit solution to the corresponding Poisson equation. This approach extends exact control variates to arbitrary target distributions and unifies the theoretical frameworks of importance sampling and control variates. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art samplers and existing control variate techniques on both synthetic and real-world posterior distributions.

Abstract

Markov chain Monte Carlo (MCMC) methods are the standard tool for computing expectations under complex probability distributions. Control variates reduce the variance of the resulting estimates, but a good control variate requires solving the Poisson equation of the sampler, which rarely admits a closed-form solution. Exact solutions are available when the sampler's kernel has a known spectral decomposition on a simple reference density. In our work, we extend these solutions to general targets through a learned change of variables. A bijection, such as a normalizing flow, is trained so that the target becomes close to the reference in a latent space, and we show that Markov kernels and their Poisson solutions are transformed by any bijection. Running such samplers in the latent space then yields explicit control variates, and the estimator is consistent under mild tail conditions on the map and target. Importance sampling (IS) from the flow is the limiting case of the same construction and the control variates apply to it as well. Experiments on synthetic targets and real posteriors compare the procedure against state-of-the-art samplers and control variates.

Problem

• Markov chain Monte Carlo
• variance reduction
• control variates
• Poisson equation
• normalizing flow

Innovation

• Markov chain Monte Carlo
• Control variates
• Poisson equation
• Normalizing flow
• Variance reduction