Demystifying Monte Carlo methods in R: A guide from Metropolis–Hastings to Hamiltonian Monte Carlo with biological growth equation examples

Hamiltonian Monte Carlo (HMC) has emerged as a cutting-edge and versatile Markov Chain Monte Carlo (MCMC) method, widely adopted across various disciplines due to its superior computational efficiency compared to other MCMC techniques. Over the past few decades, HMC has gained significant traction. However, its implementation can pose challenges for practitioners, as it relies on concepts rooted in Hamiltonian dynamics from classical mechanics. Despite the development of modern Bayesian computation tools like Stan, which facilitate the application of HMC, the underlying mechanics may remain opaque to beginners. This article seeks to provide a clear and accessible introduction to HMC. We begin by reviewing the Metropolis–Hastings (MH) algorithm and its limitations, illustrated with simulated data. We then methodically explain the HMC algorithm through step-by-step simulation examples, showcasing its implementation in R software. Finally, we present a series of ecological case studies spanning a broad range of applications, including both single-species and multispecies dynamics. These studies demonstrate the implementation of HMC using the rstan package in R, applied to both simulated and real-world data. By adopting this pedagogical approach, we aim to help newcomers better understand and apply HMC to their research domains with confidence.