In sampling tasks, it is common for target distributions to be known up to a normalizing constant. However, in many situations, even evaluating the unnormalized distribution can be costly or infeasible. This issue arises in scenarios such as sampling from the Bayesian posterior for tall datasets and the `doubly-intractable' distributions. In this paper, we begin by observing that seemingly different Markov chain Monte Carlo (MCMC) algorithms, such as the exchange algorithm, PoissonMH, and TunaMH, can be unified under a simple common procedure. We then extend this procedure into a novel framework that allows the use of auxiliary variables in both the proposal and the acceptance--rejection step. Several new MCMC algorithms emerge from this framework that uses estimated gradients to guide the proposal moves. They have demonstrated significantly better performance than existing methods on both synthetic and real datasets. We also develop theory for the new framework and use it to simplify and extend results for existing algorithms. The code to reproduce the experimental results can be found at https://github.com/ywwes26/Auxiliary-MCMC.

翻译：在采样任务中，目标分布通常仅已知至归一化常数。然而，在许多情况下，即便计算未归一化分布的取值也可能代价高昂或不可行。该问题出现在诸如对大数据集的贝叶斯后验分布采样以及“双重难解”分布等场景中。本文首先观察到，看似不同的马尔可夫链蒙特卡洛（MCMC）算法（如交换算法、PoissonMH和TunaMH）可统一于一种简单通用流程。随后，我们将该流程扩展为一种新颖框架，允许在提议步骤和接受-拒绝步骤中使用辅助变量。基于此框架衍生出多种新型MCMC算法，这些算法利用估计梯度引导提议移动。在合成数据集和真实数据集上，它们展现出显著优于现有方法的性能。我们还为这一新框架建立了理论基础，并利用该理论简化并扩展了现有算法的相关结论。复现实验结果的代码可在 https://github.com/ywwes26/Auxiliary-MCMC 获取。