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 utilize estimated gradients to guide the proposal moves. They have demonstrated significantly better performance than existing methods on both synthetic and real datasets. Additionally, we develop the theory of the new framework and apply it to existing algorithms to simplify and extend their results.

翻译：在采样任务中，目标分布通常仅已知到一个归一化常数。然而，在许多情况下，即使评估未归一化的分布也可能计算成本高昂或不可行。这一问题出现在诸如从高维数据集的贝叶斯后验分布以及“双重难解”分布中采样的场景中。本文首先指出，诸如交换算法、PoissonMH 和 TunaMH 等看似不同的马尔可夫链蒙特卡洛（MCMC）算法，可以在一个简单的通用流程下得到统一。随后，我们将此流程扩展为一个新颖的框架，该框架允许在提议步骤和接受-拒绝步骤中均使用辅助变量。基于此框架衍生出若干新的 MCMC 算法，这些算法利用估计的梯度来引导提议移动。在合成数据集和真实数据集上的实验表明，其性能显著优于现有方法。此外，我们发展了新框架的理论，并将其应用于现有算法，以简化和扩展其相关结果。