带方差缩减的变换采样器

中文摘要 AI 辅助

马尔可夫链蒙特卡洛（MCMC）方法是计算复杂概率分布下期望的标准工具。控制变量可降低估计结果的方差，但良好的控制变量需要求解采样器的泊松方程，而该方程极少存在闭式解。当采样器的核在简单参考密度上具有已知谱分解时，可获得精确解。本研究通过学习变量变换，将这些解扩展到一般目标分布。训练一个双射（如归一化流），使目标在潜在空间中接近参考分布，且证明马尔可夫核及其泊松解可被任意双射变换。在潜在空间中运行此类采样器可得到显式控制变量，且在映射和目标的温和尾部条件下，估计量是一致的。从流进行重要性采样（IS）是同一构造的极限情况，控制变量也适用于它。在合成目标和真实后验上的实验，将该方法与最先进的采样器和控制变量进行了比较。

英文摘要

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.

发表机构

• University College London（伦敦大学学院）
• Google DeepMind（谷歌DeepMind）
• Athens University of Economics and Business（雅典经济与商业大学）

机构由 AI 辅助整理，请以论文原文为准。