Adaptive Parameters tuning based on energy-preserving splitting integration for Hamiltonian Monte Carlo Method

Splitting schemes are numerical integrators for Hamiltonian problems that may advantageously replace the { St\"ormer-Verlet} method within Hamiltonian Monte Carlo (HMC) methodology. However, HMC performance is very sensitive to the step size parameter; in this paper we propose a new method in the one-parameter family of second-order of splitting procedures that uses a well-fitting parameter that nullifies the expectation of the energy error for univariate and multivariate Gaussian distributions, taken as a problem-guide for more realistic situations; we also provide a new algorithm that through an adaptive choice of the $b$ parameter and the step-size ensures high sampling performance of HMC. For similar methods introduced in recent literature, by using the proposed step size selection, the splitting integration within HMC method never rejects a sample when applied to univariate and multivariate Gaussian distributions. For more general non Gaussian target distributions the proposed approach exceeds the principal especially when the adaptive choice is used. The effectiveness of the proposed is firstly tested on some benchmarks examples taken from literature. Then, we conduct experiments by considering as target distribution, the Log-Gaussian Cox process and Bayesian Logistic Regression.