Piecewise-deterministic Markov process (PDMP) samplers constitute a state of the art Markov chain Monte Carlo (MCMC) paradigm in Bayesian computation, with examples including the zig-zag and bouncy particle sampler (BPS). Recent work on the zig-zag has indicated its connection to Hamiltonian Monte Carlo, a version of the Metropolis algorithm that exploits Hamiltonian dynamics. Here we establish that, in fact, the connection between the paradigms extends far beyond the specific instance. The key lies in (1) the fact that any time-reversible deterministic dynamics provides a valid Metropolis proposal and (2) how PDMPs' characteristic velocity changes constitute an alternative to the usual acceptance-rejection. We turn this observation into a rigorous framework for constructing rejection-free Metropolis proposals based on bouncy Hamiltonian dynamics which simultaneously possess Hamiltonian-like properties and generate discontinuous trajectories similar in appearance to PDMPs. When combined with periodic refreshment of the inertia, the dynamics converge strongly to PDMP equivalents in the limit of increasingly frequent refreshment. We demonstrate the practical implications of this new paradigm, with a sampler based on a bouncy Hamiltonian dynamics closely related to the BPS. The resulting sampler exhibits competitive performance on challenging real-data posteriors involving tens of thousands of parameters.

翻译：分段确定性马尔可夫过程（PDMP）采样器构成了贝叶斯计算中先进的马尔可夫链蒙特卡洛（MCMC）范式，其典型实例包括 zig-zag 采样器和弹跳粒子采样器（BPS）。近期关于 zig-zag 的研究揭示了其与哈密顿蒙特卡洛（一种利用哈密顿动力学的 Metropolis 算法变体）的关联。本文证明，这两种范式间的联系实际上远超特定实例的范畴。其核心在于：（1）任何时间可逆的确定性动力学均可构成有效的 Metropolis 提案；（2）PDMP 特有的速度变换机制构成了对传统接受-拒绝步骤的替代方案。基于此观察，我们建立了严谨的理论框架，通过弹跳哈密顿动力学构造无拒绝的 Metropolis 提案——该动力学同时具备类哈密顿特性，并生成与 PDMP 外观相似的不连续轨迹。当结合惯量的周期性重置时，随着重置频率的不断增加，该动力学会强收敛至对应的 PDMP 等价形式。我们通过构建与 BPS 密切相关的弹跳哈密顿动力学采样器，展示了新范式的实际意义。该采样器在涉及数万个参数的复杂实际数据后验分布上表现出具有竞争力的性能。