基于速度跳跃过程的Gibbs分布精确目标化
Exact targeting of Gibbs distributions using velocity-jump processes
摘要：
本文引入并研究了一类新的速度跳跃Markov过程，它具有以下两个性质：（1）当时间步长参数向给定的Langevin动力学或Hamilton动力学方向为零时，轨迹按规律收敛; ii）过程的平稳分布总是由高斯分布的乘积精确给出，（对于速度）由任何目标对数密度，其梯度是逐点计算的，以及一些额外的显式适当的上界。该过程不表现出任何速度反射（跳跃大小可以控制），适用于“因子分解法”。我们提供了一个严格的数学证明：i）小的时间步长收敛到哈密顿/朗之万动力学，以及ii）指数快速收敛到目标分布时，适当的噪音速度。数值实现的详细说明。
英文摘要：
This work introduces and studies a new family of velocity jump Markov processes directly amenable to exact simulation with the following two properties: i) trajectories converge in law when a time-step parameter vanishes towards a given Langevin or Hamil-tonian dynamics; ii) the stationary distribution of the process is always exactly given by the product of a Gaussian (for velocities) by any target log-density whose gradient is pointwise computabe together with some additional explicit appropriate upper bound. The process does not exhibit any velocity reflections (jump sizes can be controlled) and is suitable for the 'factorization method'. We provide a rigorous mathematical proof of: i) the small time-step convergence towards Hamiltonian/Langevin dynamics, as well as ii) the exponentially fast convergence towards the target distribution when suitable noise on velocity is present. Numerical implementation is detailed and illustrated.
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