Piecewise deterministic Markov processes (PDMPs) are a class of continuous-time Markov processes that were recently used to develop a new class of Markov chain Monte Carlo algorithms. However, the implementation of the processes is challenging due to the continuous-time aspect and the necessity of integrating the rate function. Recently, Corbella, Spencer, and Roberts (2022) proposed a new algorithm to automate the implementation of the Zig-Zag sampler. However, the efficiency of the algorithm highly depends on a hyperparameter ($t_{\text{max}}$) that is fixed all along the run of the algorithm and needs preliminary runs to tune. In this work, we relax this assumption and propose a new variant of their algorithm that let this parameter change over time and automatically adapt to the target distribution. We also replace the Brent optimization algorithm by a grid-based method to compute the upper bound of the rate function. This method is more robust to the regularity of the function and gives a tighter upper bound while being quicker to compute. We also extend the algorithm to other PDMPs and provide a Python implementation of the algorithm based on JAX.

翻译：分段确定性马尔可夫过程是一类连续时间马尔可夫过程，近年来被用于开发新型马尔可夫链蒙特卡洛算法。然而，由于连续时间特性及速率函数积分的必要性，该过程的实现具有挑战性。近期，Corbella、Spencer与Roberts（2022）提出了一种自动化实现Zig-Zag采样器的新算法。但该算法的效率高度依赖于一个在算法全程运行中固定不变且需通过预运行调试的超参数（$t_{\text{max}}$）。本研究通过放宽此假设条件，提出其算法的新变体：该参数可随时间动态调整，并自动适应目标分布。同时，我们采用基于网格的方法替代Brent优化算法来计算速率函数的上界。此方法对函数正则性更具鲁棒性，在提升计算速度的同时能获得更紧凑的上界。我们还将该算法扩展至其他分段确定性马尔可夫过程，并提供了基于JAX的Python算法实现。