Predictive Coding: A Free-Energy Model of an Adaptive Mind

World & Attention

Sensory precision π₀ 1.0

Prior precision π₁ 1.0

Inference rate 2.0

Free Energy

F = 0.00

Sensory error ‖ε₀‖

0.00

Belief error ‖ε₁‖

0.00

Explained variance

Time

0.0s

How it works

This is a three-layer hierarchical predictive-coding network — the mechanism behind the "adaptive internal state" theories of machine cognition describe: a system that continuously updates a model of itself and the world to minimise surprise. Each layer i tries to predict the layer below it with weights W, producing a prediction error ε that drives inference:

pred₀ = W₁ · tanh(x₁)        ε₀ = d(t) − pred₀   (sensory layer)
pred₁ = W₂ · tanh(x₂)        ε₁ = x₁ − pred₁     (hidden layer)
F = ½ ( π₀‖ε₀‖² + π₁‖ε₁‖² )                        (variational free energy)

Belief states x₁ and x₂ perform gradient descent on F — exactly the update Karl Friston's free-energy principle and Rao & Ballard's 1999 predictive-coding model of cortex prescribe:

ẋ₁ = π₀ W₁ᵀ · [ε₀ ⊙ (1−tanh²x₁)] − π₁ε₁ − λx₁
ẋ₂ = π₁ W₂ᵀ · [ε₁ ⊙ (1−tanh²x₂)] − λx₂

Sensory precision π₀ — how much the top layers trust incoming data versus their own prior; raising it makes beliefs snap to the signal faster (mimics attention).
Prior precision π₁ — how strongly the hidden layer trusts its own top-down prediction from the belief layer above.
Inference rate — the step size of the gradient-descent update (how fast beliefs are revised).
Inject surprise — jumps the sensory signal to an unpredicted value; watch ε₀ and F spike, then fall back down as the network re-infers new beliefs that explain the surprising input away.

Node colour encodes each unit's belief (tanh activation); node glow encodes the local prediction error it is currently carrying. This is a toy but numerically real instance of the same free-energy minimisation that theories of self-modelling, adaptive "conscious-like" systems point to as a candidate mechanism.

Low prediction error High prediction error

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