Automated Techniques for Efficient Sampling of Piecewise-Deterministic Markov Processes

Abstract: 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.