The free energy principle is a theoretical framework suggesting that the brain reduces surprise or uncertainty by making predictions based on internal models and updating them using sensory input. It highlights the brain's objective of aligning its internal model with the external world to enhance prediction accuracy. This principle integrates Bayesian inference with active inference, where actions are guided by predictions and sensory feedback refines them. It has wide-ranging implications for comprehending brain function, perception, and action.^[1]

Overview

In biophysics and cognitive science, the free energy principle is a mathematical principle describing a formal account of the representational capacities of physical systems: that is, why things that exist look as if they track properties of the systems to which they are coupled.^[2]

It establishes that the dynamics of physical systems minimise a quantity known as surprisal (which is just the negative log probability of some outcome); or equivalently, its variational upper bound, called free energy. The principle is used especially in Bayesian approaches to brain function, but also some approaches to artificial intelligence; it is formally related to variational Bayesian methods and was originally introduced by Karl Friston as an explanation for embodied perception-action loops in neuroscience.^[3]

The free energy principle models the behaviour of systems that are distinct from, but coupled to, another system (e.g., an embedding environment), where the degrees of freedom that implement the interface between the two systems is known as a Markov blanket. More formally, the free energy principle says that if a system has a "particular partition" (i.e., into particles, with their Markov blankets), then subsets of that system will track the statistical structure of other subsets (which are known as internal and external states or paths of a system).

The free energy principle is based on the Bayesian idea of the brain as an “inference engine.” Under the free energy principle, systems pursue paths of least surprise, or equivalently, minimize the difference between predictions based on their model of the world and their sense and associated perception. This difference is quantified by variational free energy and is minimized by continuous correction of the world model of the system, or by making the world more like the predictions of the system. By actively changing the world to make it closer to the expected state, systems can also minimize the free energy of the system. Friston assumes this to be the principle of all biological reaction.^[4] Friston also believes his principle applies to mental disorders as well as to artificial intelligence. AI implementations based on the active inference principle have shown advantages over other methods.^[4]

The free energy principle is a mathematical principle of information physics: much like the principle of maximum entropy or the principle of least action, it is true on mathematical grounds. To attempt to falsify the free energy principle is a category mistake, akin to trying to falsify calculus by making empirical observations. (One cannot invalidate a mathematical theory in this way; instead, one would need to derive a formal contradiction from the theory.) In a 2018 interview, Friston explained what it entails for the free energy principle to not be subject to falsification: "I think it is useful to make a fundamental distinction at this point—that we can appeal to later. The distinction is between a state and process theory; i.e., the difference between a normative principle that things may or may not conform to, and a process theory or hypothesis about how that principle is realized. Under this distinction, the free energy principle stands in stark distinction to things like predictive coding and the Bayesian brain hypothesis. This is because the free energy principle is what it is — a principle. Like Hamilton's principle of stationary action, it cannot be falsified. It cannot be disproven. In fact, there’s not much you can do with it, unless you ask whether measurable systems conform to the principle. On the other hand, hypotheses that the brain performs some form of Bayesian inference or predictive coding are what they are—hypotheses. These hypotheses may or may not be supported by empirical evidence."^[5] There are many examples of these hypotheses being supported by empirical evidence.^[6]

Background

The notion that self-organising biological systems – like a cell or brain – can be understood as minimising variational free energy is based upon Helmholtz’s work on unconscious inference^[7] and subsequent treatments in psychology^[8] and machine learning.^[9] Variational free energy is a function of observations and a probability density over their hidden causes. This variational density is defined in relation to a probabilistic model that generates predicted observations from hypothesized causes. In this setting, free energy provides an approximation to Bayesian model evidence.^[10] Therefore, its minimisation can be seen as a Bayesian inference process. When a system actively makes observations to minimise free energy, it implicitly performs active inference and maximises the evidence for its model of the world.

However, free energy is also an upper bound on the self-information of outcomes, where the long-term average of surprise is entropy. This means that if a system acts to minimise free energy, it will implicitly place an upper bound on the entropy of the outcomes – or sensory states – it samples.^[11][12]

Relationship to other theories

Active inference is closely related to the good regulator theorem^[13] and related accounts of self-organisation,^[14][15] such as self-assembly, pattern formation, autopoiesis^[16] and practopoiesis.^[17] It addresses the themes considered in cybernetics, synergetics^[18] and embodied cognition. Because free energy can be expressed as the expected energy of observations under the variational density minus its entropy, it is also related to the maximum entropy principle.^[19] Finally, because the time average of energy is action, the principle of minimum variational free energy is a principle of least action. Active inference allowing for scale invariance has also been applied to other theories and domains. For instance, it has been applied to sociology,^{[20][21][22][23]} linguistics and communication,^[24][25][26] semiotics,^[27][28] and epidemiology ^[29] among others.

Negative free energy is formally equivalent to the evidence lower bound, which is commonly used in machine learning to train generative models, such as variational autoencoders.

Action and perception

Active inference applies the techniques of approximate Bayesian inference to infer the causes of sensory data from a 'generative' model of how that data is caused and then uses these inferences to guide action. Bayes' rule characterizes the probabilistically optimal inversion of such a causal model, but applying it is typically computationally intractable, leading to the use of approximate methods. In active inference, the leading class of such approximate methods are variational methods, for both practical and theoretical reasons: practical, as they often lead to simple inference procedures; and theoretical, because they are related to fundamental physical principles, as discussed above.

These variational methods proceed by minimizing an upper bound on the divergence between the Bayes-optimal inference (or 'posterior') and its approximation according to the method. This upper bound is known as the free energy, and we can accordingly characterize perception as the minimization of the free energy with respect to inbound sensory information, and action as the minimization of the same free energy with respect to outbound action information. This holistic dual optimization is characteristic of active inference, and the free energy principle is the hypothesis that all systems which perceive and act can be characterized in this way.

In order to exemplify the mechanics of active inference via the free energy principle, a generative model must be specified, and this typically involves a collection of probability density functions which together characterize the causal model. One such specification is as follows. The system is modelled as inhabiting a state space ${\displaystyle X}$, in the sense that its states form the points of this space. The state space is then factorized according to ${\displaystyle X=\Psi \times S\times A\times R}$, where ${\displaystyle \Psi }$ is the space of 'external' states that are 'hidden' from the agent (in the sense of not being directly perceived or accessible), ${\displaystyle S}$ is the space of sensory states that are directly perceived by the agent, ${\displaystyle A}$ is the space of the agent's possible actions, and ${\displaystyle R}$ is a space of 'internal' states that are private to the agent.

Keeping with the Figure 1, note that in the following the ${\displaystyle {\dot {\psi }},\psi ,s,a}$ and ${\displaystyle \mu }$ are functions of (continuous) time ${\displaystyle t}$. The generative model is the specification of the following density functions:

• A sensory model, ${\displaystyle p_{S}:S\times \Psi \times A\to \mathbb {R} }$, often written as ${\displaystyle p_{S}(s\mid \psi ,a)}$, characterizing the likelihood of sensory data given external states and actions;
• a stochastic model of the environmental dynamics, ${\displaystyle p_{\Psi }:\Psi \times \Psi \times A\to \mathbb {R} }$, often written ${\displaystyle p_{\Psi }({\dot {\psi }}\mid \psi ,a)}$, characterizing how the external states are expected by the agent to evolve over time ${\displaystyle t}$, given the agent's actions;
• an action model, ${\displaystyle p_{A}:A\times R\times S\to \mathbb {R} }$, written ${\displaystyle p_{A}(a\mid \mu ,s)}$, characterizing how the agent's actions depend upon its internal states and sensory data; and
• an internal model, ${\displaystyle p_{R}:R\times S\to \mathbb {R} }$, written ${\displaystyle p_{R}(\mu \mid s)}$, characterizing how the agent's internal states depend upon its sensory data.

These density functions determine the factors of a "joint model", which represents the complete specification of the generative model, and which can be written as

${\displaystyle p_{\text{Bayes}}({\dot {\psi }},s,a,\mu \mid \psi )=p_{S}(s\mid \psi ,a)p_{\Psi }({\dot {\psi }}\mid \psi ,a)p_{A}(a\mid \mu ,s)p_{R}(\mu \mid s)}$.

Bayes' rule then determines the "posterior density" ${\displaystyle p({\dot {\psi }}|s,a,\mu ,\psi )}$, which expresses a probabilistically optimal belief about the external state ${\displaystyle \psi }$ given the preceding state and the agent's actions, sensory signals, and internal states. Since computing ${\displaystyle p_{\text{Bayes}}}$ is computationally intractable, the free energy principle asserts the existence of a "variational density" ${\displaystyle q({\dot {\psi }}|s,a,\mu ,\psi )}$, where ${\displaystyle q}$ is an approximation to ${\displaystyle p_{\text{Bayes}}}$. One then defines the free energy as

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