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），该方法作为技术工具对均匀和极切片抽样进行插值与外推。通过考察$k$-PSS的Wasserstein收缩率与谱间隙，我们获得了关于该方法在不同目标分布下性能的强定量结论。由于$k$-PSS包含均匀和极切片抽样作为特例，我们的结果显著推进了对这两种方法的理论理解。特别地，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度的现实估计，另一方面证明了其对近任意多元t分布的收敛速度估计。此外，我们的结果表明，对于重尾分布，极切片抽样具有与维度无关的优良性能，而均匀切片抽样则遭受相当严重的维度诅咒。