Free-Energy Principle (FEP)

• Free-Energy Principle (FEP) is a framework that defines how adaptive systems, like brains and artificial agents, maintain homeostasis by minimizing variational free energy.
• It unifies diverse processes such as perception, learning, and action under a common mechanism based on gradient descent and active inference.
• The principle has far-reaching applications in cognitive neuroscience, AI, control engineering, and biophysical modeling, linking neural dynamics to behavior.

The Free-Energy Principle (FEP) is a theoretical framework that formalizes how adaptive systems—including brains, biological organisms, and certain artificial agents—maintain their existence and homeostasis in uncertain environments. According to the FEP, such systems minimize a quantity called variational free energy, which upper-bounds sensory “surprise” (negative log-evidence) and thereby unifies processes of inference, learning, perception, and action within a consistent mathematical structure. The FEP has been applied across multiple domains, including cognitive neuroscience, AI, control engineering, machine learning, biophysics, and even quantum information theory.

1. Mathematical Foundation and General Formulation

At the core of the FEP is the assertion that an agent maintains an internal generative model (or “recognition density”) over the hidden causes of its sensory inputs and uses this model to minimize free energy—a functional that measures the divergence between its beliefs and the probabilistic structure of the environment. Formally, for internal states $b$, sensory data $s$, and world states $\psi$, the continuous and discrete formulations are:

$$F(s, b) = \int q(\psi|b) \ln \frac{q(\psi|b)}{p(\psi, s|m)} d\psi$$

$$F(b', b, s, a) = \sum_{\psi'} q(\psi'|b') \ln \frac{q(\psi'|b')}{p(\psi', s|b, a)}$$

Here, $q(\psi|b)$ is the agent's belief (recognition) density about latent causes, and $p(\psi, s|m)$ is the generative density encoding the agent's modeled joint probability over sensory data and causes given model $m$ (McGregor et al., 2015, Buckley et al., 2017).

The FEP achieves its unifying role by enabling the recasting of perception, action, and learning as processes of free energy minimization, often implemented via gradient descent. Under this paradigm, perception approximates Bayesian filtering, while action and policy selection ("active inference") induce expected sensory outcomes consistent with prior or target beliefs (Buckley et al., 2017, Laar et al., 2019).

The FEP also admits hierarchical and dynamic extensions using generalized coordinates (e.g., $(\mu, \mu', \mu'')$) to accommodate state variables and their derivatives, allowing it to model temporal and hierarchical aspects of brain function (Buckley et al., 2017).

2. Active Inference: Unified Perception and Action

Active inference refers to the process wherein the minimization of free energy is used not only to update beliefs about the world (perception) but also to select actions that shape future sensory data to conform with the agent's priors, or "expectances." In a minimal agent model, the process is:

• Internal beliefs (encoded as a softmax over brain states $b$) are updated via gradient descent on free energy:

$$b'_j \leftarrow b'_j - \eta \frac{\partial F(b',b,s,a)}{\partial b'_j}$$

where $\eta$ is a learning rate (McGregor et al., 2015).

• Actions $a$ are selected to minimize free energy with respect to predicted sensory consequences:

$a^* = \arg\min_a F(\sim s(a), b)$

• The same free energy function governs both belief updating and action selection, demonstrating the absence of a strict perception-action dichotomy (McGregor et al., 2015, Buckley et al., 2017).

In agent-based simulations, such as those implemented in discrete (bit-based) environments or one-dimensional “thermostat” worlds, the FEP-driven agent performs approximate Bayesian inference online, explaining current sensory data and acting to achieve preferred sensory outcomes (McGregor et al., 2015, Buckley et al., 2017).

3. Biophysical and Synaptic Implementations

The FEP's mathematical structure extends down to the level of synaptic dynamics. At the synapse, the postsynaptic membrane potential, synaptic weights, and the stochastic release process are modeled as latent variables governed by stochastic differential equations:

• Synaptic weight adjusts via a locally computable gradient (e.g., triplet STDP rule):

$$\Delta w = W_\mathrm{LTP}(\Delta t_1, \Delta t_2) - \left(\frac{1}{2} + w\right) W_\mathrm{LTD}(\Delta t_1, \Delta t_2) + \frac{1}{2w}$$

where $W_\mathrm{LTP}$ and $W_\mathrm{LTD}$ are functions of pre- and post-synaptic spike timing (Kappel et al., 2021).

• The learning rule naturally incorporates uncertainty at the synapse, driving the emergence of homeostasis and representing uncertainty in ambiguous network states (Kappel et al., 2021).

More generally, Bayesian mechanics under the FEP can be formulated using Onsager–Machlup theory, leading to continuous-time least-action principles and associated Hamiltonian dynamics for synaptic and neuronal variables (Kim, 3 Oct 2024). The resulting canonical equations capture the evolution of states, weights, and their associated “error momenta” in phase space:

$$\dot{\mu} = (1/m_\mu) p_\mu - \gamma_\mu (\mu - \mu_d) + w s$$

$\dot{w} = (1/m_w) p_w - \gamma_w (w - w_d) + s \mu$

$\dot{p}_\mu = \gamma_\mu p_\mu - s p_w$

$\dot{p}_w = \gamma_w p_w - s p_\mu$

where $p_\mu$, $p_w$ are momenta conjugate to $\mu$ and $w$ (Kim, 3 Oct 2024).

4. Hierarchies, Uncertainty, and Model Structure

A critical feature of the FEP is the assumption of a Markov blanket, which enables a functional separation between internal and external states, mediated by active and sensory states (Buckley et al., 2017, Millidge et al., 2021, Costa, 15 Oct 2024):

• The Markov blanket ensures that, conditioned on the blanket, internal and external states are independent: $p(y, x | b) = p(y|b)p(x|b)$.
• In Gaussian models, this corresponds to a block structure in the precision matrix (zero off-diagonal blocks between internal and external states conditioned on the blanket) (Aguilera et al., 2021).
• The free energy decomposes as:

  $$F(b,\mu) = D_\mathrm{KL}[q_\mu(\eta) \,\|\, p(\eta|b)] - \log p(b, \mu)$$

• This separation allows hierarchical generative models, with each level handling separate timescales and abstraction layers (e.g., from synapse to cortex).

A significant point of caution is that the neat statistical separation required by the FEP is only strictly realized under a narrow set of symmetric coupling conditions; in most real biological systems with perception-action asymmetries or strong recurrent couplings, these conditions are violated, complicating direct application (Aguilera et al., 2021, Millidge et al., 2021).

5. Practical Applications: Estimation, Control, and Machine Learning

The FEP provides a unifying variational framework for estimation and optimal control:

• In estimation and control problems, the FEP yields a free energy function that trades off information-theoretic surprise and predictive (cumulative) control cost:

$$\mathcal{F}_t[q] = D[q || f_t] = \int q(s) \log \left( \frac{q(s)}{f_t(s)} \right) ds$$

where $f_t$ merges the generative model with control cost via an exponential cost prior (Laar et al., 2019).

• For linear Gaussian dynamics with quadratic cost, the FEP-derived controller recovers the classical LQG solution under deterministic and vanishing-cost limits. For finite cost weightings, the FEP-based controller is more conservative, reflecting explicit incorporation of process noise (Laar et al., 2019).
• In reinforcement learning and imitation learning, unifying both as active inference within the FEP reduces exploration costs, improves sample efficiency, and can outperform pure behavioral cloning and model-based RL baselines, especially in sparse-reward settings (Ogishima et al., 2021).
• Neural network implementations, such as the Helmholtz machine, can be directly interpreted as FEP systems, with wake-sleep training and active sampling corresponding to free energy minimization and active inference, respectively. Active inference can be exploited to deform the data distribution in favor of confirming the model's latent space, increasing performance (Liu, 2023).
• Embodied perception systems leverage the FEP to unite sensorimotor control and perception (e.g., sequential gaze selection in visual categorization), with confirmation bias and attentional strategies emerging naturally from free energy minimization under resource constraints (Esaki et al., 2020).

6. Extensions, Limitations, and Theoretical Connections

The FEP has been extended to quantum information theory, suggesting that generic quantum systems can be modeled as observers that minimize Bayesian prediction error at an informational boundary (“holographic screen”), and that under error minimization, the system's generative model converges to unitary evolution (Fields et al., 2021). This suggests a link between the FEP and foundational physics (e.g., unitarity).

In recent research, the FEP has been used to explain deviations from pure scale-free network structure in real-world systems as consequences of constrained agent-level information processing, predicting characteristic “knee” shapes in degree distributions when saturation and optimality regimes are considered (Williams et al., 18 Feb 2025).

Critical limitations of the FEP include:

• The stringent requirements for a clean Markov blanket and symmetrical perception-action decomposition, which may not hold in complex, cyclic, or strongly coupled biological systems (Aguilera et al., 2021, Millidge et al., 2021).
• The assumption—only rarely fulfilled in practice—that the system dynamics align with instantaneous gradient descent on expected free energy, neglecting history and nonlocal effects (Millidge et al., 2021, Aguilera et al., 2021).
• The regular use of the Laplace approximation (Gaussian variational densities), which may limit generality and accuracy for highly multimodal or non-Gaussian systems.

7. Role in Cognition, Neuroscience, and AI

The FEP is widely posited as a unifying theory for brain function:

• Predictive coding models, which formalize how cortical circuits minimize prediction error, are tightly linked to the FEP’s hierarchical, variational inference structure (Buckley et al., 2017, Millidge et al., 2021).
• In neuroscience, parameter and hyperparameter learning under the FEP has been used to explain both synaptic plasticity and rapid neural activity, with timescale separation emerging naturally from the formulation (Buckley et al., 2017).
• The principle of “self-evidencing” (matching internal models to experienced input) underlies a normative view of adaptation, homeostasis, and cognition (Friston et al., 2022, Costa, 15 Oct 2024).
• In machine learning, FEP-derived objectives underpin modern variational autoencoders and related deep generative models, often providing intrinsic motivation for exploration or learning (Liu, 2023).

The FEP has been shown to generate, under energy and morphological constraints, neuromorphic architectures that support hierarchical, topologically organized computation, linking information geometry and quantum computation to cognitive biology (Fields et al., 2022). In linguistics, principles of efficient computation (minimal search and Kolmogorov complexity minimization) in syntax have been interpreted as direct realizations of free energy minimization (Murphy et al., 2022).

8. Summary Table: Principal Mathematical Objects

Mathematical Object	Role in FEP	Representative Formula
Variational Free Energy (F)	Upper bound on surprise, objective function	$$F = \int q(\psi) \ln \frac{q(\psi)}{p(\psi,s)} d\psi$$
Recognition Density ($q$)	Agent’s beliefs over hidden causes	$q(\psi|b)$ or $q(\theta)$
Generative Density ($p$)	Agent's model of joint sensory–latent structure	$p(\psi, s|m)$
Gradient Descent	Belief updating/action selection mechanism	$$b_{j}' \leftarrow b_{j}' - \eta \frac{\partial F}{\partial b_{j}'}$$
Markov Blanket	Statistical boundary for internal–external separation	$p(y, x|b) = p(y|b)p(x|b)$
Hamiltonian/Lagrangian	Dynamic formulation in phase space	$\mathcal{L}, \mathcal{H}$ from Onsager–Machlup or least action (Kim, 3 Oct 2024)

The FEP continues to evolve as a foundational theory, both in neuroscience and in broader biophysical and cognitive domains. Its capacity to bridge thermodynamics, variational inference, and adaptive behavior makes it a central object of theoretical and empirical investigation; however, its empirical generality, range of applicability, and mathematical underpinnings remain areas of ongoing rigorous scrutiny (Aguilera et al., 2021, Millidge et al., 2021, Costa, 15 Oct 2024).