  Many living and complex systems exhibit second order emergent dynamics.
Limited experimental access to the configurational degrees of freedom results
in data that appears to be generated by a non-Markovian process. This poses a
challenge in the quantitative reconstruction of the model from experimental
data, even in the simple case of equilibrium Langevin dynamics of Hamiltonian
systems. We develop a novel Bayesian inference approach to learn the parameters
of such stochastic effective models from discrete finite length trajectories.
We first discuss the failure of naive inference approaches based on the
estimation of derivatives through finite differences, regardless of the time
resolution and the length of the sampled trajectories. We then derive, adopting
higher order discretization schemes, maximum likelihood estimators for the
model parameters that provide excellent results even with moderately long
trajectories. We apply our method to second order models of collective motion
and show that our results also hold in the presence of interactions.
