Free energy principle

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 and subsequent treatments in psychology and machine learning. 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. Therefore, its minimization can be seen as a Bayesian inference process. When a system actively makes observations to minimize free energy, it implicitly performs active inference and maximizes 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 minimize free energy, it will implicitly place an upper bound on the entropy of the outcomes – or sensory states – it samples.

Relationship to other theories

Active inference is closely related to the good regulator theorem and related accounts of self-organisation, such as self-assembly, pattern formation, autopoiesis and practopoiesis. It addresses the themes considered in cybernetics, synergetics 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. 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, linguistics and communication, semiotics, and epidemiology 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 underlies 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 ${\displaystyle X=\mathrm{\Psi }\times S\times A\times R}$, where ${\displaystyle \mathrm{\Psi }}$ 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, ${\displaystyle p_S:S\times \mathrm{\Psi }\times A\to [0,1]}$, often written as ${\displaystyle p_S(s\mid \psi ,a)}$, characterizing the likelihood of sensory data ${\displaystyle s}$ given external states ${\displaystyle \psi }$ and agent actions ${\displaystyle a}$;
• a stochastic model of the environmental dynamics, ${\displaystyle p_{\mathrm{\Psi }}:\mathrm{\Psi }\times \mathrm{\Psi }\times A\to [0,1]}$, often written ${\displaystyle p_{\mathrm{\Psi }}(\dot{\psi }\mid \psi ,a)}$, characterizing how the external states are expected by the agent to evolve over time ${\displaystyle t}$, given the agent's actions;
• an action model, ${\displaystyle p_A:A\times R\times S\to [0,1]}$, written ${\displaystyle p_A(a\mid \mu ,s)}$, characterizing how the agent's actions depend upon its internal states ${\displaystyle \mu }$ and sensory data; and
• an internal model, ${\displaystyle p_R:R\times S\to [0,1]}$, 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

${\displaystyle p(\dot{\psi },s,a,\mu \mid \psi )=p_S(s\mid \psi ,a)p_{\mathrm{\Psi }}(\dot{\psi }\mid \psi ,a)p_A(a\mid \mu ,s)p_R(\mu \mid s)}$.

Bayes' rule then determines the "posterior density" ${\displaystyle p_{\text{Bayes}}(\dot{\psi }|s,a,\mu ,\psi )}$, which expresses a probabilistically optimal belief about the external state ${\displaystyle \dot{\psi }}$ given the preceding state ${\displaystyle \psi }$ 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

${\displaystyle \begin{array}{cc}\underset{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}-\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}}{\underbrace{F(\mu ,a;s)}}& =\underset{\text{expected energy}}{\underbrace{{\mathbb{E}}_{q(\dot{\psi })}[-\log p(\dot{\psi },s,a,\mu \mid \psi )]}}-\underset{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}{\underbrace{\mathbb{H}[q(\dot{\psi }\mid s,a,\mu ,\psi )]}}\\ & =\underset{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{e}}{\underbrace{-\log p(s)}}+\underset{\mathrm{d}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}{\underbrace{\mathbb{K}\mathbb{L}[q(\dot{\psi }\mid s,a,\mu ,\psi )\parallel p_{\text{Bayes}}(\dot{\psi }\mid s,a,\mu ,\psi )]}}\\ & \ge \underset{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{e}}{\underbrace{-\log p(s)}}\end{array}}$

and defines action and perception as the joint optimization problem

${\displaystyle \begin{array}{cc}{\mu }^{\ast }& =\underset{\mu }{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\{F(\mu ,a;s)\}\\ a^{\ast }& =\underset{a}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\{F({\mu }^{\ast },a;s)\}\end{array}}$

where the internal states ${\displaystyle \mu }$ are typically taken to encode the parameters of the 'variational' density ${\displaystyle q}$ and hence the agent's "best guess" about the posterior belief over ${\displaystyle \mathrm{\Psi }}$. Note that the free energy is also an upper bound on a measure of the agent's (marginal, or average) sensory surprise, and hence free energy minimization is often motivated by the minimization of surprise.

Free energy minimisation

Self-organisation

Free energy minimisation has been proposed as a hallmark of self-organising systems when cast as random dynamical systems. This formulation rests on a Markov blanket (comprising action and sensory states) that separates internal and external states. If internal states and action minimise free energy, then they place an upper bound on the entropy of sensory states:

${\displaystyle \underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}\underset{\text{free-action}}{\underbrace{{\int }_0^{T}F(s(t),\mu (t))dt}}\ge \underset{T\to \mathrm{\infty }}{lim}\frac{1}{T}{\int }_0^T\underset{\text{surprise}}{\underbrace{-\log p(s(t)\mid m)}}dt=H[p(s\mid m)]}$

This is because – under ergodic assumptions – the long-term average of surprise is entropy. This bound resists a natural tendency to disorder – of the sort associated with the second law of thermodynamics and the fluctuation theorem. However, formulating a unifying principle for the life sciences in terms of concepts from statistical physics, such as random dynamical system, non-equilibrium steady state and ergodicity, places substantial constraints on the theoretical and empirical study of biological systems with the risk of obscuring all features that make biological systems interesting kinds of self-organizing systems.

Bayesian inference

All Bayesian inference can be cast in terms of free energy minimisation. When free energy is minimised with respect to internal states, the Kullback–Leibler divergence between the variational and posterior density over hidden states is minimised. This corresponds to approximate Bayesian inference – when the form of the variational density is fixed – and exact Bayesian inference otherwise. Free energy minimisation therefore provides a generic description of Bayesian inference and filtering (e.g., Kalman filtering). It is also used in Bayesian model selection, where free energy can be usefully decomposed into complexity and accuracy:

${\displaystyle \underset{\text{free-energy}}{\underbrace{F(s,\mu )}}=\underset{\text{complexity}}{\underbrace{D_{\mathrm{K}\mathrm{L}}[q(\psi \mid \mu )\parallel p(\psi \mid m)]}}-\underset{\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{y}}{\underbrace{E_q[\log p(s\mid \psi ,m)]}}}$

Models with minimum free energy provide an accurate explanation of data, under complexity costs; cf. Occam's razor and more formal treatments of computational costs. Here, complexity is the divergence between the variational density and prior beliefs about hidden states (i.e., the effective degrees of freedom used to explain the data).

Thermodynamics

Variational free energy is an information-theoretic functional and is distinct from thermodynamic (Helmholtz) free energy. However, the complexity term of variational free energy shares the same fixed point as Helmholtz free energy (under the assumption the system is thermodynamically closed but not isolated). This is because if sensory perturbations are suspended (for a suitably long period of time), complexity is minimised (because accuracy can be neglected). At this point, the system is at equilibrium and internal states minimise Helmholtz free energy, by the principle of minimum energy.

Information theory

Free energy minimisation is equivalent to maximising the mutual information between sensory states and internal states that parameterise the variational density (for a fixed entropy variational density). This relates free energy minimization to the principle of minimum redundancy.

Neuroscience

Free energy minimisation provides a useful way to formulate normative (Bayes optimal) models of neuronal inference and learning under uncertainty and therefore subscribes to the Bayesian brain hypothesis. The neuronal processes described by free energy minimisation depend on the nature of hidden states: ${\displaystyle \mathrm{\Psi }=X\times \mathrm{\Theta }\times \mathrm{\Pi }}$ that can comprise time-dependent variables, time-invariant parameters and the precision (inverse variance or temperature) of random fluctuations. Minimising variables, parameters, and precision correspond to inference, learning, and the encoding of uncertainty, respectively.

Perceptual inference and categorisation

Free energy minimisation formalises the notion of unconscious inference in perception and provides a normative (Bayesian) theory of neuronal processing. The associated process theory of neuronal dynamics is based on minimising free energy through gradient descent. This corresponds to generalised Bayesian filtering (where ~ denotes a variable in generalised coordinates of motion and ${\displaystyle D}$ is a derivative matrix operator):

${\displaystyle \dot{\tilde{\mu }}=D\tilde{\mu }-{\mathrm{\partial }}_{\mu }F(s,\mu ){|}_{\mu =\tilde{\mu }}}$

Usually, the generative models that define free energy are non-linear and hierarchical (like cortical hierarchies in the brain). Special cases of generalised filtering include Kalman filtering, which is formally equivalent to predictive coding – a popular metaphor for message passing in the brain. Under hierarchical models, predictive coding involves the recurrent exchange of ascending (bottom-up) prediction errors and descending (top-down) predictions that is consistent with the anatomy and physiology of sensory and motor systems.

Perceptual learning and memory

In predictive coding, optimising model parameters through a gradient descent on the time integral of free energy (free action) reduces to associative or Hebbian plasticity and is associated with synaptic plasticity in the brain.

Perceptual precision, attention and salience

Optimizing the precision parameters corresponds to optimizing the gain of prediction errors (cf., Kalman gain). In neuronally plausible implementations of predictive coding, this corresponds to optimizing the excitability of superficial pyramidal cells and has been interpreted in terms of attentional gain.

With regard to the top-down vs. bottom-up controversy, which has been addressed as a major open problem of attention, a computational model has succeeded in illustrating the circular nature of the interplay between top-down and bottom-up mechanisms. Using an established emergent model of attention, namely SAIM, the authors proposed a model called PE-SAIM, which, in contrast to the standard version, approaches selective attention from a top-down position. The model takes into account the transmission of prediction errors to the same level or a level above, in order to minimise the energy function that indicates the difference between the data and its cause, or, in other words, between the generative model and the posterior. To increase validity, they also incorporated neural competition between stimuli into their model. A notable feature of this model is the reformulation of the free energy function only in terms of prediction errors during task performance:

${\displaystyle {\displaystyle \frac{\mathrm{\partial }{E}^{total}(Y^{VP},X^{SN},x^{CN},y^{KN})}{\mathrm{\partial }{y}_{mn}^{SN}}}=x_{mn}^{CN}-b^{CN}{\epsilon }_{nm}^{CN}+b^{CN}\sum _k({\epsilon }_{knm}^{KN})}$

where ${\displaystyle E^{total}}$ is the total energy function of the neural networks entail, and ${\displaystyle {\epsilon }_{knm}^{KN}}$ is the prediction error between the generative model (prior) and posterior changing over time. Comparing the two models reveals a notable similarity between their respective results while also highlighting a remarkable discrepancy, whereby – in the standard version of the SAIM – the model's focus is mainly upon the excitatory connections, whereas in the PE-SAIM, the inhibitory connections are leveraged to make an inference. The model has also proved to be fit to predict the EEG and fMRI data drawn from human experiments with high precision. In the same vein, Yahya et al. also applied the free energy principle to propose a computational model for template matching in covert selective visual attention that mostly relies on SAIM. According to this study, the total free energy of the whole state-space is reached by inserting top-down signals in the original neural networks, whereby we derive a dynamical system comprising both feed-forward and backward prediction error.

Active inference

When gradient descent is applied to action ${\displaystyle \dot{a}=-{\mathrm{\partial }}_{a}F(s,\tilde{\mu })}$, motor control can be understood in terms of classical reflex arcs that are engaged by descending (corticospinal) predictions. This provides a formalism that generalizes the equilibrium point solution – to the degrees of freedom problem – to movement trajectories.

Optimal control

Active inference is related to optimal control by replacing value or cost-to-go functions with prior beliefs about state transitions or flow. This exploits the close connection between Bayesian filtering and the solution to the Bellman equation. However, active inference starts with (priors over) flow ${\displaystyle f=\mathrm{\Gamma }\cdot \mathrm{\nabla }V+\mathrm{\nabla }\times W}$ that are specified with scalar ${\displaystyle V(x)}$ and vector ${\displaystyle W(x)}$ value functions of state space (cf., the Helmholtz decomposition). Here, ${\displaystyle \mathrm{\Gamma }}$ is the amplitude of random fluctuations and cost is ${\displaystyle c(x)=f\cdot \mathrm{\nabla }V+\mathrm{\nabla }\cdot \mathrm{\Gamma }\cdot V}$. The priors over flow ${\displaystyle p(\tilde{x}\mid m)}$ induce a prior over states ${\displaystyle p(x\mid m)=\exp (V(x))}$ that is the solution to the appropriate forward Kolmogorov equations. In contrast, optimal control optimises the flow, given a cost function, under the assumption that ${\displaystyle W=0}$ (i.e., the flow is curl free or has detailed balance). Usually, this entails solving backward Kolmogorov equations.

Optimal decision (game) theory

Optimal decision problems (usually formulated as partially observable Markov decision processes) are treated within active inference by absorbing utility functions into prior beliefs. In this setting, states that have a high utility (low cost) are states an agent expects to occupy. By equipping the generative model with hidden states that model control, policies (control sequences) that minimise variational free energy lead to high utility states.

Neurobiologically, neuromodulators such as dopamine are considered to report the precision of prediction errors by modulating the gain of principal cells encoding prediction error. This is closely related to – but formally distinct from – the role of dopamine in reporting prediction errors per se and related computational accounts.

Cognitive neuroscience

Active inference has been used to address a range of issues in cognitive neuroscience, brain function and neuropsychiatry, including action observation, mirror neurons, saccades and visual search, eye movements, sleep, illusions, attention, action selection, consciousness, hysteria and psychosis. Explanations of action in active inference often depend on the idea that the brain has 'stubborn predictions' that it cannot update, leading to actions that cause these predictions to come true.