It is easy to get lost among all probability densities, random variables, and functionals of the Free Energy Principle (Active Inference Framework). This post is a summary of the relevant ontology.
The Objective Physics
The Free Energy Principle begins with a strictly physical, deductive ontology of existence. It defines what is required for a system to resist entropic dissipation and maintain structural integrity.
The Global State Space (\(x\))
The universe is modeled as a random dynamical system. The state of this system is defined by a joint state vector \(x\), containing all interacting variables.
The Markov Blanket Partition (\(\eta, o, a, \mu\))
To distinguish an entity from its environment, the state space \(x\) must possess a topological boundary, a Markov Blanket, that mediates causal influence. This boundary partitions the system into four sets of states:
- \(\eta\) (External states): Objective physical reality outside the boundary.
- \(o\) (Sensory states): The incoming boundary; caused by \(\eta\) and influencing \(\mu\).
- \(a\) (Active states; Actions): The outgoing boundary; caused by \(\mu\) and influencing \(\eta\).
- \(\mu\) (Internal states): Objective physical reality inside the boundary (e.g., neuronal configurations). They are conditionally independent of \(\eta\) given \(o\) and \(a\).
Non-Equilibrium Steady State (NESS) & Attracting Set (\(p(\eta, o, a, \mu)\))
The objective, global probability density function of the joint state space. To exist as a persistent entity means the joint state vector is restricted to a bounded, recurring trajectory. The region of phase space where this density is maximized constitutes the system’s attracting set. Divergence from this set entails a phase transition (dissolution).
Objective Marginal Density (\(p(o)\))
The actual, physical distribution of sensory states produced when the system successfully remains within its NESS. It is the objective requirement for the phenotype’s structural viability:
$$p(o) = \iiint p(\eta, o, a, \mu) \, d\eta \, da \, d\mu$$
The Subjective Computation (The Generative Model)
Because the Markov blanket conditionally isolates \(\mu\) from \(\eta\), the internal states cannot compute actions using the objective variables. They must construct a statistical architecture to infer causality and evaluate policies.
Hidden States (\(s\))
The subjective, hypothesized causes of sensory data. They are computational constructs distinct from the true external physical states (\(\eta\)). Called mental states in earlier posts.
The Generative Model (\(p(o, s, \pi, \phi)\))
The internal causal mapping parameterized by the physical configuration of \(\mu\). It is structurally composed of distinct probabilistic components, operating under modifiable parameters (\(\phi\)) updated via learning.
Empirical Prior (\(p(s)\) or \(p(s_t|s_{t-1}, \pi)\))
The descriptive component of the generative model. It dictates the context-dependent transition dynamics—how the agent expects the hidden states of the world to evolve over time, independent of its own survival requirements. Priors can be updated quickly. For instance, if the recognition distribution peaks at “rain”, then the prior is quickly set to peak at “rain” as it is likely that it will rain a minut from now if it rains now. Priors are, contrary to folk psychology interpretation, rather fluid.
Likelihood (\(p(o|s)\))
The descriptive mapping that translates hypothesized hidden states into expected sensory observations.
Prior Preference / Target Distribution (\(\tilde{p}(o)\) or \(\tilde{p}(s)\))
The normative component of the generative model. It is the internal parameterization of the objective attracting set. It dictates the required sensory or hidden states the phenotype must secure to persist. In reality the preference distribution is conditional on the current state so that it becomes \(\tilde p(o \mid s)q(s)\). If \(q(s)\) for instance peaks at state “hungry” when perceptual inference is completed, then the preference distribution would peak for observations of food.
Recognition Density (\(q(s)\), \(q(s|o)\), or \(q(s|o, \mu)\))
The agent’s current probabilistic belief about the hidden states, parameterized directly by \(\mu\) in response to incoming sensory data \(o\).
The Key Functionals
The internal states \(\mu\) operate strictly to minimize bounding functionals, creating the macroscopic equivalence of Bayesian inference and goal-directed action.
Variational Free Energy (\(\mathcal{F}\))
The functional minimized to optimize perception and learning. It mathematically bounds sensory surprise (\(-\ln p(o)\)). By physically reconfiguring \(\mu\) to minimize \(\mathcal{F}\), the system forces the recognition density \(q(s)\) to approximate the true posterior, and updates the structural parameters \(\phi\) of the generative model.
$$\mathcal{F} = \mathbb{E}_{q(s)}[\ln q(s) – \ln p(o, s)]$$
Expected Free Energy (\(\mathcal{G}\))
The functional minimized to select policies (\(\pi\)) and drive active states (\(a\)). It is computed prospectively by projecting the generative model forward in time. It forces the selection of actions that simultaneously resolve uncertainty (Ambiguity) and minimize the divergence between the predicted consequences of the policy and the encoded prior preferences (Risk).
$$\mathcal{G}(\pi) \approx \underbrace{\mathbb{E}_{q(s|\pi)}[H(p(o|s))]}_{\text{Ambiguity}} + \underbrace{D_{KL}(q(o|\pi) \parallel \tilde{p}(o))}_{\text{Risk}}$$