The free energy principle—a precis

17 October 2023

1. Introduction

Here, we present the free energy principle (FEP) in a simple manner to a broad audience. Alongside other cornerstones of mathematical physics—for instance, variational principles such as the principles of stationary action or the principle of maximum entropy—the FEP serves as the basis for a new class of mechanics or mechanical theories (in the manner that the principle of stationary action leads to classical mechanics, or the principle of maximum entropy leads to statistical mechanics). This new physics has been called Bayesian mechanics [1], and comprises tools that allow us to model the time evolution of things or particles within a system that are coupled to, but distinct from, other such particles [2]. More specifically, it allows us to partition a system of interest into “particles” or “things” that can be distinguished from other things [3]. This coupling is sometimes discussed in terms of probabilistic “beliefs” that things encode about each other; in the sense that coupled systems carry information about each other—because they are coupled. The FEP allows us to specify mathematically the time evolution of a coupled random dynamical system, in a way that links the evolution of the system and that of its “beliefs” over time (or belief updating: changing your mind in light of new evidence or information).

We begin by presenting an overview of the FEP. The FEP rests on sparse coupling, that is, the idea that “things” can be defined in terms of the absence of direct influence between subsets of a system (that can be partitioned into things or particles). Markov blankets formalize a specific kind of sparseness or separation between things, which we take to be definitional of things per se under the FEP. We then discuss the multi-scale aspect of the FEP: namely, that this self-similar pattern repeats at every scale at which things can be observed—from rocks to rockstars, and beyond. We discuss the broader philosophical implications of this approach to thingness via sparseness. We conclude with a discussion of the integrative or unificatory implications of the FEP for the study of life and mind.

2. Information physics from first principles

The free energy principle is a mathematical principle that allows us to describe the time evolution of coupled random dynamical systems. It was proposed by neuroscientist Karl Friston in the mid-2000s, in the context of theoretical neurobiology [4].  The FEP exists at the intersection of mathematics, statistics, and physics, namely: classical, statistical, and quantum mechanics, dynamical systems theory, and information theory.  Since its inception it has been applied to a wide range of disciplines, from wet neuroscience and computational modeling [5] and nonequilibrium physics [6, 3], to psychology [7], robotics [8, 9], collective behavior [10, 11], morphogenesis [12, 13], evolutionary biology and natural selection [14], niche construction theory [15], echo chamber formation and information spread in social networks [16], and to artificial intelligence [17].

Heuristically, the FEP tells us something deep about what it means for things to be coupled to each other, but distinct from one another. It implies that any thing that exists physically—in the sense that it can be reliably reidentified over time as “the same thing,” an idea that we discuss below—will necessarily look as if it “tracks” the things to which it is coupled. Loosely speaking, the FEP says that observable things do on average what it is characteristic or typical for them to do, given what they are—and that in doing so, they end up attuning to (and indeed tuning) the statistics of the things to which they are coupled. Mathematically speaking, this “tracking” behavior is formalized as a kind of abductive inference (called variational Bayes or approximate Bayesian inference), allowing the thing to attune itself to the statistics of the environment to which it is coupled (which will typically be made up of other things).

3. Some preliminary epistemology of physics: Dynamics, mechanics, and principles

More precisely, we appeal to a distinction between dynamics, mechanics, and principles, which is fairly standard in physics [1]. Dynamics are simply formal descriptions of behavior. Mechanics (aka mechanical theories) are mathematical theories that are developed to explain the functional form or “shape” of dynamics, i.e., they explain why the dynamics are the way that we observe them to be. Principles, in turn, tell us where mechanical theories come from. More formally, mechanical theories allow us to generate dynamics, whereas principles can be used to write down mechanical theories. For instance, classical mechanics emerges from the principle of least action: we set the variation of the action to zero, and take the paths of the system that conform to that constraint, as the paths of the system through its state space.

The move from dynamics to mechanics and principles is well illustrated by the transition from Galileo and Kepler’s phenomenological formalizations to Newton’s principles-based formulations (simplifying the history somewhat). Kepler’s formal descriptions of the movements of heavenly bodies—formalized as geocentric ellipses—constitute a dynamics, according to this definition. Newton’s laws of motion explain (e.g.) why the elliptical trajectories of planets have the shape that they do.

The FEP is what it says on the tin: it is a principle, which can be applied as a modeling method for generic physical systems. Importantly, the FEP is not an empirical theory that could be subjected to verification or falsification. Rather, much like the principle of least action or the principle of maximum entropy in physics, the FEP is used to write down theories that can, in turn, be subjected to empirical verification procedures. The truth of the FEP is mathematical. Having said that, it is a striking fact that principles like the principle of least action and the FEP can be used to explain the behavior of things in the natural world (see [1], for a discussion). If the principle of least action leads to classical mechanics, the FEP, in turn, leads to Bayesian mechanics. This is a physics of the informational coupling between interacting things.

4. A theory of thingness (or “every thing”)

The perspective opened up by the FEP starts from the question of what it means to be an observable thing in our physical universe. In other words, it assumes the rest of physics and attempts to answer that question specifically in the context of physical reality. The (classical) formulation of the FEP answers the question by appealing to the separation of timescales: at some time scale, it looks as if there are things that can be reliably (re)identified, and that persist through time as what they are [3]. The FEP is thus at its very core a dynamical systems approach, focused on the time evolution of coupled random dynamical systems—which are both separate from but coupled to each other—and also on modeling the time evolution of this coupling itself [6].

In that sense, it attempts an answer to the question that is preliminary to state space modeling. Indeed, when one uses the FEP, one usually begins by specifying a temporal scale, formally doing so via the apparatus of the Langevin equation, which has the form

x˙(τ ) = f (x, τ ) + ω

This  allows  us  to  describe  the  time  evolution  or  flow  x of a system as a mixture of a deterministic component f (known as the flow or drift) and a noise component ω, which represents the phenomena that are averaged out at some time scale. Given this setup, one can traverse scales of variance, from the very fast fluctuations of unimaginably small quantum processes and fields, through to the meso-scale of living systems, to large-scale cosmological phenomena—carving out “things” at each scale, in terms of the sparseness structures or Markov blankets that emerge and dissolve as one traverses scales. The notion that the evolution of systems is driven by the interaction of nested phenomena unfolding at faster and slower timescales is ubiquitous in scientific investigation, from physics to psychology—and the idea that scientifically interesting systems are nested systems, composed of other interesting systems, is core to the FEP [18, 19].

5. Sparseness and thingness

The FEP is based on the idea that sparseness is key to thingness. The main idea is that thingness is defined in terms of what is not connected to what. Think of a box containing an idealized gas, as opposed to a box containing a rock. In a gas, we have strongly mixing dynamics: any molecule in the gas could find itself arbitrarily connected to any other, such that no persistent “thing” can be identified in the box. By contrast, a rock does not mix with or dissipate into its environment—at least, not over the temporal scale that we can reliably reidentify it as being the rock that it is. In other words, the rock exists as a rock (i.e., as the thing that it is) because it is disconnected from the rest of the system in a specific way.

We said earlier that the FEP says that if a physical thing exists, then it will look as if it is tracking the things to which it is coupled. More precisely, then, the FEP states that if a physical system instantiates the right kind of sparseness, then subsets of the system (i.e., particles or things) will appear to track each other. The sparseness in play is formalized as a Markov blanket. (Note that, terminologically, we use the word “system” to refer to the whole random dynamical system, and the word “thing” or “particle” to refer to Markov blanketed subsets of that system.)

Figure 1: Markov blankets and sparseness. The Markov blanket captures the dependence structure of the dynamics or flow of a sparsely coupled random dynamical system. Note that the equations of motion show that internal states are only influenced by blanket states and other internal states; and vice versa for external states. Likewise note that sensory states are defined as those that are affected by external states and affect internal states (thereby mediating their vicarious interactions) but that are not affected by internal states; and vice versa for active states.

The Markov blanket is important because it is the way that we capture the sparse dependence structure of the system: where dependence means that one random variable quantifiably predicts the flow or dynamics of another. The blanket is defined as the set of states conditioned upon which states internal to a particle are rendered independent of states external to the particle. With the blanket so defined, the FEP says that if a system contains a Markov blanket, then states internal to the boundary will look as if they encode a kind of belief about what lies beyond the boundary, i.e., a Bayesian belief or conditional probability distribution, parameterised by internal states.

Figure 2: The free energy principle: A technical diagram. The free energy principle (FEP) says that if the generative model (or dependence structure) of a random dynamical system contains a Markov blanket, then it will look as if internal states track the statistics of external states across the boundary. Technically, these beliefs are the parameters of an approximate Bayesian or variational probability density defined over all the states of the system. Note that external states are only labeled as external relative to other particles and their internal states; see [20].

Equipped with the FEP, one can now simulate or model self-organization in terms of (Bayesian) belief updating. In other words, in the same way one can use classical mechanics to model, design, and build physical artifacts (e.g., bridges and planes), one can use Bayesian mechanics to model, design, and build things that evince “sentient” behavior (e.g., cells and minds)—where “sentient” means “responsive to sensory stimuli” (as in [21]). This is the utility of the FEP; namely, the applications discussed above. We conclude with a brief discussion of some philosophical commitments behind these applications.

6. Strange inversions and the FEP

The philosophical implications of Bayesian mechanics under the FEP have only begun to be explored (see [1], for a discussion). A few general remarks are in order. One first implication concerns the nature of explanation under the FEP. The FEP rests on a “strange inversion” (in the spirit of [22]) of the traditional explanatory strategy deployed in the sciences of life and mind. In particular, the FEP inverts the usual manner of explaining phenomena in the life sciences [23]. Typically, we have some phenomenon that we want to explain [24, 25], and the usual procedure is to identify the set of parts, operations, and their organization which together lead to the emergence of that phenomenon. The FEP inverts this style of explanation: it does not say that organisms must minimize free energy or surprise to exist. Rather, the FEP says that if things exist, then they minimize free energy or surprise—or can be modeled as such [23, 26].

A second major philosophical implication of the FEP has to do with what it means to exist as a whole. The FEP theoretic type of explanation also constitutes an inversion of the traditional metaphysics of emergence that we have inherited from Aristotle and which continues to structure thinking about emergence in philosophy and science. This is the view that the whole is “other than” or “more than the sum of its parts,” which was popularized in modern times by British emergentists [27]. On this view, the nested or multi-level structure of physical reality is best understood as the successive emergence or supervenience of new “layers of reality” from more basal phenomena (e.g., the emergence of chemistry from physics, or the emergence of biology from chemistry). On this count, the FEP underwrites another “strange inversion.” Under the FEP, the novel capabilities of the whole are underwritten by the fact that the whole is less than the sum of its parts. Indeed, a whole can only function as an integrated whole if we remove degrees of freedom from its parts. A petrol engine, for instance, can only function if the degrees of freedom of its pistons are dramatically reduced (to movement along a single axis); and the easiest way to destroy an engine would be to introduce such new degrees of freedom. The same could be said about cancer formation in living tissues, etc. (see [28]). Of course, the phenomena that emerge from interactions between the constrained components are vastly different from their unconstrained counterparts and may appear new or unpredictable. However, mathematically speaking, this emergent behavior is made possible for the system because degrees of freedom have been removed from the parts. (Note that this sparseness-based perspective on thingness is closely related to discussions of the “closure of constraints” in the enactive approach [29, 30]—indeed, we know that maximizing entropy given a set of constraints is mathematically equivalent to minimizing variational free energy given a generative model [2, 1, 31]—and is closely related, but distinct from, discussions of “absential” phenomena discussed by [32].

Finally, the FEP has some deep philosophical implications for the way that we carve up the natural world. In a sense, the introduction of Bayesian mechanics echoes the move from Scholastic philosophy to classical mechanics in the late 1600s (for a related discussion in consciousness studies, see [33]). In Scholastic philosophy, following the teachings of Aristotle, the movements or behaviors of things (i.e., their dynamics) was explained by appealing to the five classical elements of nature: earth, water, air, fire, and aether. More broadly, the world was divided into the sublunary and supralunary spheres of existence. On this view, the sublunary sphere comprised the things that we know from our ordinary experience, where things are always in the process of becoming, i.e., they have a beginning, middle, and end. The behaviors of things were then explained by appealing to the properties of the classical elements. For instance, the earth element is cold and dry, and things that are of that element tend to fall. The supralunary sphere denoted the behavior of celestial bodies and the like, which were of the aether element, and thereby moved in perfect circles and not subject to decay.

The conceptual revolution introduced by classical mechanics abolished the distinction between sublunary and supralunary spheres of existence, since it could be shown that the dynamics of supralunary and sublunary things were governed by the same principles. Similarly, the FEP stands in contrast to approaches that would split the sphere of mind or life from the sphere of physical phenomena, such as some versions of the enactive approach (e.g., [34]). The FEP eschews all such distinctions and embraces a physics of thingness that ranges from subatomic particles to galaxy clusters—and every kind of thing in between. Importantly, the resulting philosophical perspective is not physicalist reductionism (a reduction of causal efficacy to “mere” physics)—but rather, a deep commitment to anti-reductionism, emphasizing the causal contributions of things at every scale to the overall dynamics of the nested system.

Acknowledgements. Thanks to Mahault Albarracin, Charles Bakker, Axel Constant, Karl Friston, Alex Kiefer, Brennan Klein, and Dalton Sakthivadivel for valuable feedback on drafts of this paper.


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