A comparative review of Monte Carlo algorithms with numerical SDE approaches for mixed effect dynamical models

Abstract

Stochastic differential equation mixed-effects models (SDEMEMs) provide a flexible framework for studying time-evolving processesgoverned by stochastic dynamics. Analytical inference is often intractable due to incomplete observations, inter-individual variability, repeated measurements, and measurement error. This work focuses on Bayesian inference for latent states and model parameters in SDEMEMs using numerical discretization and Monte Carlo methodologies. We implement three discretization schemes—Euler--Maruyama (EM), the modified diffusion bridge (MDB), and the residual diffusion bridge (RDB)—combined with Markov Chain Monte Carlo (MCMC), pseudo-marginal, and Sequential Monte Carlo (SMC) approaches. In particular, we investigate Individual Augmentation (IA), the Metropolis-adjusted Langevin algorithm (MALA), and Hamiltonian Monte Carlo (HMC). The proposed methods are evaluated using both simulated and real datasets, including a bivariate Ornstein--Uhlenbeck (OU) process and orange tree growth data. The results indicate that correlated pseudo-marginal methods improve sampling efficiency by increasing the effective sample size (ESS). Gradient-based methods, particularly HMC, generally achieve higher acceptance rates and larger ESS values than IA-based approaches, although at a higher computational cost due to repeated gradient evaluations. Among the discretization schemes, MDB and RDB improve proposal conditioning and acceptance rates relative to EM, whereas EM and MDB provide lower computational cost than RDB. These results highlight practical trade-offs between computational efficiency, sampling performance, and implementation complexity in Bayesian inference for SDEMEMs.