Abstract
The free energy principle (FEP) states that any dynamical system can be interpreted as performing Bayesian inference upon its surrounding environment. Although, in theory, the FEP applies to a wide variety of systems, there has been almost no direct exploration or demonstration of the principle in concrete systems. 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 stochastic 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.
Highlights
During the last decade, the ‘free energy principle’ (FEP) has become an influential framework which aims to provide a grand theory promoting a Bayesian interpretation of living systems [1, 2, 3]
The FEP states that any self-organizing system equipped with a Markov blanket – a statistical separation between internal and external states – can be interpreted as performing Bayesian inference upon the surrounding environment, such that its internal states come to encode probabilistic beliefs about the external environment [3, 4]
The latest formulation of the free energy principle [3, 4] states that, in any dynamical system equipped with a Markov blanket, the flow of internal states can be construed as a gradient ascent on Bayesian model evidence
Summary
The ‘free energy principle’ (FEP) has become an influential framework which aims to provide a grand theory promoting a Bayesian interpretation of living systems [1, 2, 3]. It implies that the dynamics of any pair of coupled systems, under specific conditions about the interaction of internal and external states, can be described as one system trying to statistically infer the states of the second system (cf an agent and its environment) This claim licenses an interpretation of the agent as performing a basic kind of Bayesian inference and encoding beliefs about the surrounding environment [5, 6]. Papers on the topic doubles every few years, Fig. 1), we believe it is imperative to ground and test the foundations of the theory in concrete models to assess the generality and validity of its claims To this end, we explore a class of systems defined by stochastic linear differential equations, under a weak-coupling assumption. A sufficient condition for a Markov blanket of variable x
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