Free Energy Principle I: the variational free-energy boundResearch Paper
## 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