There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution.

翻译：关于动态实现的哈密顿蒙特卡洛（HMC）方法（如无U型转向采样器（NUTS））在众多具有挑战性的推断问题中取得成功的实证证据十分充分，但有关其行为的理论结果却相对匮乏。本文旨在填补这一空白。具体而言，我们考虑一类被称为动态HMC的通用MCMC算法。我们证明这一通用框架将NUTS作为特例包含在内，并由此推导出目标分布的不变性。其次，我们建立了NUTS具有不可约性和非周期性的条件，从而得出其遍历性的推论。在与HMC现有条件相似的条件下，我们还证明NUTS具有几何遍历性。最后，我们改进了HMC的现有收敛性结果，表明在目标分布为高斯分布扰动的情况下，该方法无需对步长和蛙跳步数施加任何有界性条件即可实现遍历性。