8.7 The free-energy principle, honestly

Karl Friston’s free-energy principle (FEP) is the most ambitious framework in this space. It claims that any self-organizing system that maintains its existence over time must, mathematically, minimize a quantity called variational free energy, which approximates the negative log-evidence of its sensory observations under its generative model. Under the FEP, perception, action, learning, and even biological homeostasis are all special cases of free-energy minimization.

The variational free energy can be written, for a system with hidden states $\theta$ and observations $S$ and a generative model $p(\theta, S)$ and a recognition distribution $q(\theta)$ , as

F = \mathbb{E}_q[\log q(\theta)] - \mathbb{E}_q[\log p(\theta, S)] = \mathrm{KL}\bigl(q(\theta) \,\Vert\, p(\theta | S)\bigr) - \log p(S).

▶ Derivation: variational free energy as a bound on negative log-evidence

Suppose we want to compute the log-evidence $\log p(S)$ for our observations $S$ under a generative model $p(\theta, S) = p(S|\theta) p(\theta)$ . This is the quantity we’d like to maximize — it tells us how well our model explains the data.

Computing $\log p(S) = \log \int p(\theta, S) d\theta$ exactly requires integrating over all hidden states $\theta$ , which is intractable.

Introduce an approximate posterior $q(\theta)$ , our best current guess at $p(\theta|S)$ . Multiply and divide inside the integral by $q(\theta)$ :

\log p(S) = \log \int q(\theta) \cdot \frac{p(\theta, S)}{q(\theta)} d\theta = \log \mathbb{E}_q\!\left[\frac{p(\theta, S)}{q(\theta)}\right].

By Jensen’s inequality (log is concave), $\log \mathbb{E}[X] \geq \mathbb{E}[\log X]$ :

\log p(S) \geq \mathbb{E}_q\!\left[\log \frac{p(\theta, S)}{q(\theta)}\right] = \mathbb{E}_q[\log p(\theta, S)] - \mathbb{E}_q[\log q(\theta)].

This lower bound is the evidence lower bound (ELBO). The variational free energy is its negative:

F = -\text{ELBO} = \mathbb{E}_q[\log q(\theta)] - \mathbb{E}_q[\log p(\theta, S)].

Then $F$ is an upper bound on negative log-evidence: $F \geq -\log p(S)$ , with equality when $q(\theta) = p(\theta|S)$ .

Equivalently, $F$ can be rewritten as

F = \mathrm{KL}\bigl(q(\theta)\, \Vert\, p(\theta|S)\bigr) - \log p(S),

making it the Kullback-Leibler divergence between the approximate and true posteriors, minus log-evidence. Minimizing $F$ does two things: (i) brings $q$ closer to the true posterior, (ii) maximizes log-evidence (model fit).

In the FEP, the brain is hypothesized to perform inference and learning by minimizing variational free energy. Perception minimizes $F$ over $q$ (refining the posterior); action minimizes $F$ over observations $S$ (changing the world to match predictions). ∎

The first term is the Kullback-Leibler divergence between the recognition distribution and the true posterior — a measure of how good the brain’s approximation is. The second term is the negative log-evidence. Minimizing $F$ is mathematically equivalent to (a) bringing the recognition distribution closer to the true posterior, and (b) maximizing the evidence the model has for its observations.

This is, with the right interpretation, a perfectly reasonable framework. It contains predictive coding as a special case (a Gaussian approximation under particular assumptions). It is also the subject of substantial criticism, ranging from claims of unfalsifiability (Bowers & Davis 2012, in Psychological Bulletin) to concerns about how the principle is operationalized in different contexts. I will not adjudicate this controversy here. The principle is productive — it has generated useful predictions and useful theory — and it is also not yet a settled brain theory in the way that, say, the wave equation is a settled equation. Treat it as a research program, not a result.