Bayesian Inference for Inertial Langevin Dynamics

Many living and complex systems exhibit second order emergent dynamics. Examples range from flocks and swarms, to bacterial propulsion, worm dynamics and cell migration experiments. A working description for these systems is given by stochastic underdamped equations. Limited experimental access to the inertial degrees of freedom poses a challenge in the quantitative reconstruction of the model, even in the case of equilibrium passive systems, as it makes the data appear to be generated by a non-Markovian process. We developed a novel analytical Bayesian approach to learn the parameters of such stochastic effective models from discrete finite-length trajectories. Naive approaches based on the estimation of derivatives through finite differences fail, yielding biased estimators regardless of the time resolution and length of the sampled trajectories. We derived, adopting a higher-order discretization, maximum-likelihood parameter estimators that provide worthy results even with moderately long trajectories. The method applies to a wide range of models, including nonlinear and nonstationary processes as well as to second-order models of collective motion, showing that reliable parameter estimators can be built also in the presence of interactions and for out-of-equilibrium systems.