We propose a new class of Markov chain Monte Carlo methods, called $k$-polar slice sampling ($k$-PSS), as a technical tool that interpolates between and extrapolates beyond uniform and polar slice sampling. By examining Wasserstein contraction rates and spectral gaps of $k$-PSS, we obtain strong quantitative results regarding its performance for different kinds of target distributions. Because $k$-PSS contains uniform and polar slice sampling as special cases, our results significantly advance the theoretical understanding of both of these methods. In particular, we prove realistic estimates of the convergence rates of uniform slice sampling for arbitrary multivariate Gaussian distributions on the one hand, and near-arbitrary multivariate t-distributions on the other. Furthermore, our results suggest that for heavy-tailed distributions, polar slice sampling performs dimension-independently well, whereas uniform slice sampling suffers a rather strong curse of dimensionality.

翻译：我们提出了一类新的马尔可夫链蒙特卡洛方法，称为$k$-极切片抽样（$k$-PSS），作为在均匀切片抽样与极坐标切片抽样之间进行插值和外推的技术工具。通过考察$k$-PSS的Wasserstein收缩率和谱隙，我们获得了关于其针对不同类型目标分布性能的强定量结果。由于$k$-PSS将均匀切片抽样和极坐标切片抽样作为特例包含在内，我们的成果显著推进了对这两种方法的理论理解。具体而言，我们一方面证明了均匀切片抽样对任意多元高斯分布的收敛速率具有现实估计，另一方面也证明了其对近乎任意多元t分布的收敛速率。此外，我们的结果表明，对于重尾分布，极坐标切片抽样表现出与维度无关的良好性能，而均匀切片抽样则受到相当严重的维度灾难影响。