This paper discusses the irreducibility and geometric ergodicity of the
Hamiltonian Monte Carlo (HMC) algorithm. We consider cases where the number of
steps of the symplectic integrator is either fixed or random. Under mild
conditions on the potential $\F$ associated with target distribution $\pi$, we
first show that the Markov kernel associated to the HMC algorithm is
irreducible and recurrent. Under more stringent conditions, we then establish
that the Markov kernel is Harris recurrent. Finally, we provide verifiable
conditions on $\F$ under which the HMC sampler is geometrically ergodic.

本文讨论了 Hamiltonian Monte Carlo 算法的不可约性和几何遍历性，考虑了较为宽松和严格的条件，得出了 HMC 采样器具有几何遍历性的可验证条件。

Hamiltonian Monte Carlo 收敛性研究

On the convergence of Hamiltonian Monte Carlo

Hamiltonian Monte Carlo can provide powerful inference in complex statistical
problems, but ultimately its performance is sensitive to various tuning
parameters. In this paper we use the underlying geometry of Hamiltonian Monte
Carlo to construct a universal optimization criteria for tuning the step size
of the symplectic integrator crucial to any implementation of the algorithm as
well as diagnostics to monitor for any signs of invalidity. An immediate
outcome of this result is that the suggested target average acceptance
probability of 0.651 can be relaxed to $0.6 \lesssim a \lesssim 0.9$ with
larger values more robust in practice.

该论文提出了一个基于 Hamiltonian Monte Carlo 的优化准则，旨在调整算法的关键性步长值，并通过诊断监测来确保其有效性，从而使接受概率调整到 0.6≤a≤0.9 之间。