  The aim of this paper is to introduce a field of study that has emerged over
the last decade called Bayesian mechanics. Bayesian mechanics is a
probabilistic mechanics, comprising tools that enable us to model systems
endowed with a particular partition (i.e., into particles), where the internal
states (or the trajectories of internal states) of a particular system encode
the parameters of beliefs about external states (or their trajectories). These
tools allow us to write down mechanical theories for systems that look as if
they are estimating posterior probability distributions over the causes of
their sensory states. This provides a formal language for modelling the
constraints, forces, potentials, and other quantities determining the dynamics
of such systems, especially as they entail dynamics on a space of beliefs
(i.e., on a statistical manifold). Here, we will review the state of the art in
the literature on the free energy principle, distinguishing between three ways
in which Bayesian mechanics has been applied to particular systems (i.e.,
path-tracking, mode-tracking, and mode-matching). We go on to examine a duality
between the free energy principle and the constrained maximum entropy
principle, both of which lie at the heart of Bayesian mechanics, and discuss
its implications.
