  The free energy principle (FEP) states that any dynamical system can be
interpreted as performing Bayesian inference upon its surrounding environment.
In this work, we examine in depth the assumptions required to derive the FEP in
the simplest possible set of systems -- weakly-coupled non-equilibrium linear
stochastic systems. Specifically, we explore (i) how general the requirements
imposed on the statistical structure of a system are and (ii) how informative
the FEP is about the behaviour of such systems. We discover that two
requirements of the FEP -- the Markov blanket condition (i.e. a statistical
boundary precluding direct coupling between internal and external states) and
stringent restrictions on its solenoidal flows (i.e. tendencies driving a
system out of equilibrium) -- are only valid for a very narrow space of
parameters. Suitable systems require an absence of perception-action
asymmetries that is highly unusual for living systems interacting with an
environment. More importantly, we observe that a mathematically central step in
the argument, connecting the behaviour of a system to variational inference,
relies on an implicit equivalence between the dynamics of the average states of
a system with the average of the dynamics of those states. This equivalence
does not hold in general even for linear systems, since it requires an
effective decoupling from the system's history of interactions. These
observations are critical for evaluating the generality and applicability of
the FEP and indicate the existence of significant problems of the theory in its
current form. These issues make the FEP, as it stands, not straightforwardly
applicable to the simple linear systems studied here and suggest that more
development is needed before the theory could be applied to the kind of complex
systems that describe living and cognitive processes.
