Transformed Samplers with Variance Reduction
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.
Related
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BibTeX
@misc{liu2026transformed,
title = {{Transformed Samplers with Variance Reduction}},
author = {Siran Liu and Michalis Tisias and Petros Dellaportas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10870},
eprint = {2610.10870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10870},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Transformed Samplers with Variance Reduction
AU - Siran Liu
AU - Michalis Tisias
AU - Petros Dellaportas
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10870
UR - https://arxiv.org/abs/2610.10870
AB - 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.
ER -