Free Energy Principle II: expected free energy, Markov blankets, Gaussian variational free energy, and Bayesian model reduction

Motivation

Mission Free Energy Principle I published the core variational step of the free energy principle (FEP): whatever recognition density a system carries, the posterior-form variational free energy never undercuts the data's surprisal, the bound is exact at the Bayesian posterior, and equality characterizes the posterior. That mission's shared finite substrate — normalized finite laws, finite kernels, entropy/cross-entropy/KL, and the finite generative model — now exists as platform definitions in the namespace FreeEnergyPrinciple.

The Free Energy Principle II mission formalizes the four structures the FEP literature builds on top of that core, all already machine-checked in the source repository fep_lean / fep_formal (Active Inference Institute):

• Expected free energy — the policy-selection functional of active inference: what a course of action is expected to cost in preference divergence and what it is expected to reveal. Its canonical decomposition [Friston et al. 2017] splits $G$ into risk (pragmatic divergence of predicted outcomes from preferences) plus ambiguity (expected entropy of outcomes given latent states), with epistemic value fixing the sign.
• Markov blankets — the partition that makes a self-organizing system statable: internal states are conditionally independent of external states given the sensory-active blanket. The source development proves this at the level of Mathlib's native conditional distributions, not as a finite mutual-information proxy.
• Gaussian variational free energy — the closed-form instantiation of the FEP-I bound for the exact scalar Gaussian filter, where the native Gaussian KL is exactly the squared mean error over twice the posterior variance.
• Bayesian model reduction — model comparison by Bayes factors: posterior odds equal prior odds times the likelihood ratio, the multiplicative update applied whenever a reduced model is compared against the model it was reduced from [Friston & Penny 2011].

Timeline of the mathematical content this mission formalizes:

• 2006/2010 — Friston's free energy principle: variational free energy as the quantity a self-organizing system minimizes (formalized in FEP-I).
• 2011 — Friston & Penny, Post hoc Bayesian model selection: Bayesian model reduction — evidence of reduced models evaluated by the free-energy difference; comparison by Bayes factors.
• 2015 — Friston, Rigoli, Sengupta, Pezzulo — the Markov-blanket partition (sensory/active states) as the geometry of the FEP.
• 2017 — Friston, FitzGerald, Rigoli, Schwartenbeck, Pezzulo, Active inference: a process theory: expected free energy $G(\pi) = \text{risk} + \text{ambiguity}$ drives policy selection.
• 2022 — Parr, Pezzulo, Friston, Active Inference (MIT Press): Gaussian treatments of filtering and the posterior-form free energy as the working equations.
• 2026 — fep_formal (Active Inference Institute): a machine-checked Lean 4 catalogue of 155 FEP topics compiled with zero proof holes against a pinned Mathlib. This mission transcribes the proved modules behind expected free energy, native Markov blankets, the scalar Gaussian filter/VFE, and Bayesian model reduction onto the platform.

Setting

Two carriers, both fully machine-checked in the source repository:

• Finite (reusing FEP-I's published substrate). Laws are normalized real mass functions on finite types; kernels are normalized rows. This mission's expected-free-energy, model-reduction, and Markov-blanket families import the published Definitions.Def_fep_finite_laws, Def_fep_finite_information, and Def_fep_generative_model — no substrate is re-published. Zero-mass atoms are handled by the same totalized conventions as FEP-I: entropy uses Real.negMulLog (so $0\log 0 = 0$ exactly), KL is the nonnegative klFun integrand, and division premises are explicit.
• Native Gaussian (self-contained on Mathlib). The Gaussian family is Mathlib's own gaussianReal/gaussianPDF at a fixed strictly positive variance; the scalar OU prediction and the closed filter update give the posterior mean/variance the recognition family varies over. The native KL between two family members is exactly $\frac{(\mu_1-\mu_2)^2}{2v}$ — proved against Mathlib's log-likelihood-ratio definition.

The four definition items of this mission package exactly these carriers:

• Def_fep2_expected_free_energy — the predicted state-outcome joint, preference risk, likelihood ambiguity, epistemic value, pragmatic cost, expected free energy (epistemic sign fixed by definition), the full-support contract, and the marginal/product/conditional-entropy/mutual-information lemmas the decomposition needs. Imports FEP-I.
• Def_fep2_gaussian_vfe — fixed-variance Gaussian family with its exact KL, scalar OU parameters, the exact scalar Gaussian filter (prediction, observation kernel, gain, closed posterior, evidence law), evidence surprisal, and the posterior-form Gaussian variational free energy. Self-contained.
• Def_fep2_bayesian_model_reduction — posterior odds, Bayes factor, and the model-odds update $odds \leftarrow odds \times Z_f/Z_r$, with totalized division boundaries kept explicit. Imports FEP-I.
• Def_fep2_native_blanket — the static blanket factorization, the Dirac-mass embedding of finite laws into native measures, blanket/internal/external coordinates, the conditional-pair kernel, and the marginal/composition identifications the independence proof needs. Imports FEP-I.

Formalization targets

Goal: expected free energy decomposes into risk plus ambiguity

For every finite generative model, every policy $\pi$, and every model with full support:

$$G[\pi] \;=\; \mathrm{KL}\big(P(o\mid\pi)\,\|\,C\big) \;+\; \sum_s P(s\mid\pi)\,H\big(A[\cdot\mid s]\big).$$

The epistemic-value sign is fixed by definition ($G[\pi] =$ pragmatic cost $-$ epistemic value); the decomposition follows from two entropy identities: epistemic value $I(s;o\mid\pi)$ is predicted outcome entropy minus ambiguity, and risk is cross-entropy minus the same entropy (Gibbs' inequality under full reference support). Nonnegativity of $G$ follows as a corollary — but the decomposition, not the bound, is the target.

Gaussian variational free energy in closed form

For the exact scalar Gaussian filter, the posterior-form variational free energy at recognition mean $\mu$ is

$$F[\mu] = \frac{(\mu - m^*)^2}{2v^*} + S(o),$$

the exact fixed-variance Gaussian KL (the recognition-to-posterior gap) plus the density-relative evidence surprisal. Equality with the surprisal holds exactly at the posterior mean — the Gaussian analogue of FEP-I's exactness theorem.

Odds recursion of Bayesian model reduction

Bayes' rule in odds form: at positive evidence,

$$\frac{P(h_f\mid e)}{P(h_r\mid e)} = \frac{P(h_f)}{P(h_r)}\cdot\frac{P(e\mid h_f)}{P(e\mid h_r)},$$

with the reference prior mass and reference likelihood as exact division premises. The multiplicative Bayes-factor structure (topic fep-120: factorized evidence ratios multiply; sequential model-odds updates agree with one update by product evidence) is available from the same definition layer as a further target.

Native Markov blanket conditional independence

The embedded static blanket factorization satisfies Mathlib's native CondIndepFun predicate: internal coordinates are conditionally independent of external coordinates given the blanket coordinate. The result is obtained by identifying the authored finite conditional kernels with Mathlib conditional distributions (the embedding preserves marginals and joints exactly on discrete carriers) — not by a finite mutual-information argument.

Significance

These four results are the load-bearing extensions of FEP-I's bound: expected free energy converts the variational principle into a theory of action selection; Markov blankets make "internal states" and "external states" well-defined relative to a blanket, which is what lets the FEP talk about self-organizing systems at all; the Gaussian filter is the tractable regime in which the variational machinery becomes the Kalman update; and Bayesian model reduction is the learning/comparison step that updates structure, not just parameters.

Formalizing them. All four families are proved with zero proof holes in the source repository, against a pinned Mathlib; the definition layer here is faithful (same carriers, same totalized conventions, same support contracts made explicit) and every item below compiles locally against the platform environment. The mission's value is reusable community infrastructure: the definition items publish the EFE layer, the Gaussian filter, the odds layer, and the native-blanket embedding in the shared namespace FreeEnergyPrinciple, so later missions (policy trees, collective inference, predictive coding) can import them instead of re-deriving. Status honesty: all eight items below are formalized and machine-checked locally against the platform environment; each is an open problem on the platform only in the sense that no proof has yet been submitted to it.

Difficulty

• The EFE decomposition looks like an algebraic rearrangement but the sign conventions are load-bearing: the epistemic value enters $G$ with a minus sign, and the two helper identities (epistemic value = outcome entropy $-$ ambiguity; risk = cross-entropy $-$ outcome entropy) both hold only under the full-support contract, which the definition makes explicit rather than hiding in a carrier.
• The Gaussian identity requires the exact native KL between Gaussian laws — the proof goes through Mathlib's log-likelihood-ratio definition and the Gaussian first moment — and the closed-form update's positivity (positive prediction variance, positive innovation variance) is what makes the recognition family genuine rather than degenerate.
• The odds recursion is a field-simp identity, but the premises are the point: a plausible rendering that hides division by zero behind totalized division changes the statement.
• The blanket theorem is the most intricate item: it must transport a finite factorization through the Dirac-mass embedding into Mathlib's conditional-distribution machinery, with nonemptiness premises for the conditional distributions to exist. A "proof" via finite mutual information would prove something weaker than the source.

Formalization scope

Committed conventions of this mission's Lean development:

• The expected-free-energy and model-reduction families reuse the published Free Energy Principle I finite substrate (namespace FreeEnergyPrinciple, definitions Def_fep_finite_laws, Def_fep_finite_information, Def_fep_generative_model); this mission adds definition items Def_fep2_expected_free_energy, Def_fep2_gaussian_vfe, Def_fep2_bayesian_model_reduction, and Def_fep2_native_blanket, all in the same namespace.
• The Gaussian family is deliberately native: Mathlib gaussianReal/gaussianPDF, no finite substrate, no manifold geometry, no singular (zero-variance) branch.
• Totalized real division boundaries (zero evidence, zero reference mass) are stated, never silently absorbed.
• The natural-gradient / dynamic-flow layer of the source's Gaussian module (natural gradient flow, strict descent away from the posterior) is deliberately left out of this mission and is a natural extension target; likewise the row-wise dynamical blanket theorem (every authored factorized transition row preserves the native blanket conditional independence), which follows directly from the static theorem via the source's nextStaticModel construction.
• Contributions welcome: the epistemic/pragmatic ENNReal balance (catalogue topic fep-021) onto this substrate, the treewise EFE decomposition (fep-133), Bayes-factor multiplicativity (fep-120), and blanket nonvacuity witnesses.

Selected references

• K. Friston, A free energy principle for the brain, Journal of Physiology (Paris) 100 (2006) 70–87. https://doi.org/10.1016/j.jphysparis.2006.10.001
• K. Friston, The free-energy principle: a unified brain theory?, Nature Reviews Neuroscience 11 (2010) 127–138. https://doi.org/10.1038/nrn2787
• K. Friston & W. Penny, Post hoc Bayesian model selection, NeuroImage 56 (2011) 2089–2099. https://doi.org/10.1016/j.neuroimage.2011.03.062
• K. Friston, T. FitzGerald, F. Rigoli, P. Schwartenbeck, G. Pezzulo, Active inference: a process theory, Neural Computation 29 (2017) 1–49. https://doi.org/10.1162/neco_a_00912
• T. Parr, G. Pezzulo, K. J. Friston, Active Inference: The Free Energy Principle in Mind, Brain, and Behavior, MIT Press (2022). https://mitpress.mit.edu/9780262045354/active-inference/
• D. A. Friedman, fep_formal: Towards Lean 4 Formalization of the Free Energy Principle (v1.2.0), Active Inference Institute (2026), the formal source of truth for this mission. https://github.com/ActiveInferenceInstitute/fep_formal
• D. A. Friedman, Towards Lean 4 Formalization of the Free Energy Principle: AI-Driven Theorem Sketching and Verification for Active Inference and Bayesian Mechanics, Active Inference Journal (2026). https://doi.org/10.5281/zenodo.19699233

Motivation

The free energy principle (FEP) proposes that a self-organizing system — a brain, an organism, an agent — persists by minimizing one quantity: the variational free energy of its sensory states under an internal generative model. Introduced by Karl Friston as a principle of brain function [Friston 2006] and stated in its unified form [Friston 2010], the principle makes a precise mathematical claim at its core: whatever internal state estimate the system holds, the free energy of incoming data is never below the data's surprisal (negative log marginal likelihood), and the excess is exactly the Kullback–Leibler divergence between the system's recognition density and the Bayesian posterior implied by the model. Active inference extends the same functional from perception to action and planning [Friston et al. 2017], and the same bound is known in machine learning as the evidence lower bound (ELBO) of variational inference [Parr et al. 2022].

Timeline of the mathematical content this mission formalizes:

• 2006 — Friston, A free energy principle for the brain (J. Physiol. Paris 100): the bound stated for perception as variational inference on a generative model.
• 2010 — Friston, The free-energy principle: a unified brain theory? (Nat. Rev. Neurosci. 11, 127–138): free energy as an upper bound on surprisal, presented as the core of a unified account.
• 2017 — Friston, FitzGerald, Rigoli, Schwartenbeck, Pezzulo, Active inference: a process theory (Neural Comput. 29(1), 1–49): the same functional drives policy selection through expected free energy.
• 2022 — Parr, Pezzulo, Friston, Active Inference (MIT Press): textbook treatment; the posterior-form identity $F = D_{\mathrm{KL}}(Q\,\|\,P(s|o)) - \log P(o)$ as the central equation.
• 2026 — fep_formal (Active Inference Institute): a machine-checked Lean 4 catalogue of 155 Free Energy Principle topics, compiled with zero proof holes against a pinned Mathlib. This mission transcribes the catalogue's core-free-energy chain — topic fep-002 and the foundation module active_inference — onto the platform, turning the first link of the FEP development into solvable community infrastructure.

Setting

Everything is finite, and laws are normalized real mass functions.

A finite law on a finite type $\alpha$ is a function $p : \alpha \to \mathbb{R}$ with $p(x) \ge 0$ for every $x$ and $\sum_x p(x) = 1$. A finite kernel from $\alpha$ to $\beta$ assigns to each $x \in \alpha$ a normalized row over $\beta$. The mission's definitions Def_fep_finite_laws and Def_fep_finite_information package these carriers with entropy, cross-entropy, and the KL divergence

$$D_{\mathrm{KL}}(p\,\|\,q) \;=\; \sum_{x} q(x)\cdot \mathrm{klFun}\!\left(\frac{p(x)}{q(x)}\right), \qquad \mathrm{klFun}(x) = x\log x + 1 - x,$$

a totalized real-valued divergence that is finite even at zero-mass atoms (the convention $0 \cdot \log 0 = 0$ via Real.negMulLog) and nonnegative on normalized laws.

A finite generative model for active inference (definition Def_fep_generative_model) over finite types $\mathsf{Policy}, \mathsf{State}, \mathsf{Outcome}$ consists of: an initial state law $P(s)$; a policy-conditioned transition kernel; a state-to-outcome likelihood kernel; a preference law over outcomes; and a policy prior. Under a policy $\pi$ the model predicts the state law $P(s \mid \pi)$ and the outcome law $P(o \mid \pi)$. A recognition density is any finite law $Q$ over states — the system's internal estimate. At an outcome $o$ with positive predicted mass, the Bayesian posterior $P(\cdot \mid o, \pi)$ is the exact finite Bayes rule. The outcome surprisal is $-\log P(o \mid \pi)$, and the posterior-form variational free energy of a recognition density $Q$ is

$$F[Q, o, \pi] \;=\; D_{\mathrm{KL}}\big(Q \,\|\, P(\cdot \mid o, \pi)\big) \;-\; \log P(o \mid \pi).$$

Formalization targets

Goal: the variational free-energy bound

For every generative model, every policy $\pi$, every outcome $o$ with $P(o\mid\pi) > 0$, and every recognition density $Q$:

$$-\log P(o \mid \pi) \;\le\; F[Q, o, \pi].$$

The recognition density $Q$ is universally quantified — the bound holds for whatever state estimate the system happens to carry.

Exactness, uniqueness, and the ELBO form

Three companions pin down the equality case, ordered weakest to strongest alongside the milestone list:

• Exactness — the Bayesian posterior attains the bound:

$$F\big[P(\cdot \mid o, \pi),\, o,\, \pi\big] = -\log P(o \mid \pi).$$

• Uniqueness — equality characterizes the posterior, with no full-support assumption:

$$F[Q, o, \pi] = -\log P(o \mid \pi) \iff Q = P(\cdot \mid o, \pi).$$

• ELBO form — negating both sides:

$$-F[Q, o, \pi] \;\le\; \log P(o \mid \pi).$$

Measure-theoretic core

Independently of the finite model, in Mathlib's nonnegative extended reals $\mathbb{R}_{\ge0} \cup \{\infty\}$, with $q$, $p$ measures on any measurable space and $s \in \mathbb{R}_{\ge0} \cup \{\infty\}$:

$$s \;\le\; s + D_{\mathrm{KL}}(q \,\|\, p),$$

the unconditional shape of the bound (topic fep-002 of the source catalogue), with the divergence taken as $\infty$ when the log-likelihood ratio is not integrable.

Significance

The result itself. This inequality is the load-bearing step of the FEP: it converts "minimize free energy" into "move recognition toward the posterior," and it is the exact statement whose continuous, dynamic, and policy-selecting extensions (expected free energy, Markov blankets, non-equilibrium thermodynamics) form the rest of the FEP literature. Without it, the principle's variational step has no mathematical content.

Formalizing it. The mathematics here is classical — Gibbs' inequality — and the source development already proves every row with no proof holes. What the mission adds is faithful, reusable infrastructure: the definitions are published as platform nodes in the shared namespace FreeEnergyPrinciple, so later missions in this programme (expected free energy and policy selection, Markov blankets, Gaussian and continuous-time variants, already proved in the source repository) can import them instead of re-deriving the substrate. Status honesty: all eight items below are formalized and machine-checked locally against the platform environment; each is an open problem on the platform only in the sense that no proof has yet been submitted to it.

Difficulty

The bound itself is a one-line consequence of KL nonnegativity — the naive idea "prove it by simp on the KL sum" is essentially right, and the source proofs are correspondingly short. The actual difficulty is boundary precision, where plausible renderings go silently wrong:

• The positivity premise $P(o \mid \pi) > 0$ is not decoration: the Bayesian posterior is defined only where the evidence has positive mass, and hiding that in a totalized division would change the statement.
• The uniqueness characterization is not a formality: at zero-mass reference atoms the logarithmic cross-entropy identity degenerates, and the proof needs the normalization lemma $D_{\mathrm{KL}}(p\,\|\,q) = 0 \leftrightarrow p = q$, which forces the recognition law's mass to zero wherever the posterior's is zero. A solver who proves the bound but states equality with a full-support hypothesis has proved something different from the source.
• The measure-theoretic core is deliberately unconditional; adding finiteness side conditions to it would weaken the source's point that $\mathbb{R}_{\ge0} \cup \{\infty\}$ absorbs the degenerate cases.

A vacuous formalization — quantifying over a single distinguished recognition law, or taking "posterior" as an arbitrary variable — would trivialize the goal; the targets below rule this out by fixing the exact finite Bayes rule and universally quantifying $Q$.

Formalization scope

Committed conventions of this mission's Lean development:

• All model carriers are finite types (Fintype); laws are $\mathbb{R}$-valued normalized mass functions; kernels are normalized rows. No measure-theoretic machinery below the finite substrate except for the measure-theoretic core milestone.
• KL is the totalized real-valued finite divergence $D_{\mathrm{KL}}(p\,\|\,q) = \sum_x q(x)\cdot \mathrm{klFun}(p(x)/q(x))$; entropy uses Real.negMulLog, so $0 \cdot \log 0 = 0$ exactly, not by exception-handling.
• The posterior is the exact finite Bayes rule FiniteKernel.posterior, taken at the explicit hypothesis $0 < P(o \mid \pi)$.
• One mission-wide namespace FreeEnergyPrinciple; definitions live in the published definition files Def_fep_finite_laws, Def_fep_finite_information, Def_fep_generative_model, and every theorem item imports them. A Free Energy Principle II mission (expected free energy) is expected to reuse the same namespace and definitions.
• The measure-theoretic core uses Mathlib's InformationTheory.klDiv in ℝ≥0∞ with no finiteness hypotheses.
• Contributions welcome: alternative measure-theoretic renderings of the core bound, the Gaussian instantiation of the same identity, and ports of the source repository's subsequent rows (Bayesian model reduction, expected free energy) onto these definitions.

Selected references

• K. Friston, A free energy principle for the brain, Journal of Physiology (Paris) 100 (2006) 70–87. https://doi.org/10.1016/j.jphysparis.2006.10.001
• K. Friston, The free-energy principle: a unified brain theory?, Nature Reviews Neuroscience 11 (2010) 127–138. https://doi.org/10.1038/nrn2787
• K. Friston, T. FitzGerald, F. Rigoli, P. Schwartenbeck, G. Pezzulo, Active inference: a process theory, Neural Computation 29 (2017) 1–49. https://doi.org/10.1162/neco_a_00912
• T. Parr, G. Pezzulo, K. J. Friston, Active Inference: The Free Energy Principle in Mind, Brain, and Behavior, MIT Press (2022). https://mitpress.mit.edu/9780262045354/active-inference/
• D. A. Friedman, fep_formal: Towards Lean 4 Formalization of the Free Energy Principle (v1.2.0), Active Inference Institute (2026), the formal source of truth for this mission. https://github.com/ActiveInferenceInstitute/fep_formal
• D. A. Friedman, Towards Lean 4 Formalization of the Free Energy Principle: AI-Driven Theorem Sketching and Verification for Active Inference and Bayesian Mechanics, Active Inference Journal (2026). https://doi.org/10.5281/zenodo.19699233