Theory: causal discovery as automata

This page summarizes the conceptual foundations behind causal-automata, drawn from the project papers Foundations of Causal Automata, Causal Automata: From Intuition to GPU Physics, and Bayesian Automata and Network Evolution (Yamagiwa, 2026). See Bibliography for citations.

Correlation vs. causation

Observational data only gives correlations. Ice-cream sales and shark attacks both rise in July; a naive model might conclude that ice cream causes sharks. Causal discovery aims to recover the mechanism — here a hidden common cause (summer temperature).

We represent mechanisms as a Directed Acyclic Graph (DAG):

  • Nodes — measured variables.

  • Directed edges — causal influence (\(A \to B\) means \(A\) causes \(B\)).

  • Acyclic — no directed cycles; time does not loop.

The software starts from a blank (or noisy) map of variables and draws arrows by evolving an adjacency state under local rules.

Independence, forks, and colliders

Dependence is measured with mutual information (MI). High MI means the variables are related; near-zero MI means independence. Conditional mutual information (CMI) \(I(X_i; X_j \mid X_k)\) asks whether knowing \(X_i\) still informs \(X_j\) after observing \(X_k\).

Fork (common cause)

Temperature
   /     \
  v       v
Ice cream  Shark attacks

\(B\) and \(C\) look dependent, but \(I(B; C \mid A) \approx 0\) once the common cause is known. Conditioning blocks the path.

Collider (V-structure)

Rain          Sprinkler
  \             /
   v           v
    Grass wet

Rain and sprinkler are independent a priori. Observing wet grass opens dependence between them. The signature

\[V_{ijk} = I(X_i; X_j \mid X_k) - I(X_i; X_j)\]

is large and positive when \(k\) is a collider (sink). The automaton uses this as a downward force — the V-well.

From Pearl to grids

Judea Pearl’s belief propagation replaced global combinatorial evaluation with local message passing: parents send causal evidence (\(\pi\)) and children send diagnostic evidence (\(\lambda\)). Cellular automata share the same engine — gather neighbors, aggregate, update — but traditionally aggregate by neighbor counts on a lattice.

Structurally, a 2D grid is an adjacency matrix with fixed degree. Belief propagation, shortest-path search, and CA differ mainly in the aggregation function (products of probabilities, min-plus distances, or active-neighbor counts).

Generative topology and Bayesian automata

Classical network analysis fixes the graph and routes information through it. A Bayesian Automaton inverts that: CA rules act on the adjacency matrix itself. Edges are born, survive, or die according to predictive power over an external dataset rather than pure geometry.

Updating every edge in parallel would create cycles. The implementation assigns each node a continuous elevation \(E \in [0, 1]\) and enforces a hard rule: edges may only flow downhill. Because arrows cannot point upward, cycles are impossible.