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（NUTS）是现代贝叶斯软件库的计算核心，但其定性与定量收敛性保证直到近期才得以确立。在其两种主要变体NUTS-mul（采用多项式采样）和NUTS-BPS（采用有偏渐进采样进行索引选择）之间的理论比较仍存在显著空白。本文通过三项贡献填补了这一空白：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NUTS-BPS在标准高斯分布上的首个混合时间结果。结果表明，NUTS-mul与NUTS-BPS展现近乎一致的定性行为，其几何遍历性取决于目标分布的尾部性质，但在收敛速率上存在定量差异。具体而言，当初始化于标准高斯测度的典型集时，两种采样器的混合时间均以$O(d^{1/4})$量级缩放（包含对数因子），其中$d$表示维度。然而，NUTS-BPS的相应常数严格小于NUTS-mul。