The No-U-Turn Sampler (NUTS) is the computational workhorse of modern Bayesian software libraries, yet its qualitative and quantitative convergence guarantees were established only recently. A significant gap remains in the theoretical comparison of its two main variants: NUTS-mul and NUTS-BPS, which use multinomial sampling and biased progressive sampling, respectively, for index selection. In this paper, we address this gap in three contributions. First, we derive the first necessary conditions for geometric ergodicity for both variants. Second, we establish the first sufficient conditions for geometric ergodicity and ergodicity for NUTS-mul. Third, we obtain the first mixing time result for NUTS-BPS on a standard Gaussian distribution. Our results show that NUTS-mul and NUTS-BPS exhibit nearly identical qualitative behavior, with geometric ergodicity depending on the tail properties of the target distribution. However, they differ quantitatively in their convergence rates. More precisely, when initialized in the typical set of the canonical Gaussian measure, the mixing times of both NUTS-mul and NUTS-BPS scale as $O(d^{1/4})$ up to logarithmic factors, where $d$ denotes the dimension. Nevertheless, the associated constants are strictly smaller for NUTS-BPS.

翻译：无回转采样器是现代贝叶斯软件库的核心计算工具，但其收敛性的定性与定量保证直到近期才得以建立。其两种主要变体——采用多项式抽样的NUTS-mul与采用有偏渐进抽样的NUTS-BPS——在索引选择机制上的理论比较仍存在显著空白。本文通过三项贡献填补这一空白：首先，我们推导了两种变体几何遍历性的首个必要条件；其次，建立了NUTS-mul几何遍历性与遍历性的首个充分条件；第三，获得了标准高斯分布下NUTS-BPS的首个混合时间结果。研究表明，NUTS-mul与NUTS-BPS在定性行为上高度一致，其几何遍历性均取决于目标分布的尾部特性。然而两者在收敛速率上存在定量差异：当初始状态位于正则高斯测度的典型集时，两种变体的混合时间均按$O(d^{1/4})$量级增长（忽略对数因子），其中$d$表示维度。但值得注意的是，NUTS-BPS所对应的常数严格小于NUTS-mul。