  The free energy principle (FEP) in the neurosciences stipulates that all
viable agents induce and minimize informational free energy in the brain to fit
their environmental niche. In this study, we continue our effort to make the
FEP a more physically principled formalism by implementing free energy
minimization based on the principle of least action. We build a Bayesian
mechanics (BM) by casting the formulation reported in the earlier publication
(Kim 2018) to considering active inference beyond passive perception. The BM is
a neural implementation of variational Bayes under the FEP in continuous time.
The resulting BM is provided as an effective Hamilton's equation of motion and
subject to the control signal arising from the brain's prediction errors at the
proprioceptive level. To demonstrate the utility of our approach, we adopt a
simple agent-based model and present a concrete numerical illustration of the
brain performing recognition dynamics by integrating BM in neural phase space.
Furthermore, we recapitulate the major theoretical architectures in the FEP by
comparing our approach with the common state-space formulations.
