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LIT-1174: Information Flows in Causal Networks

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StatusActive — read in full 2026-10-17 (NOTE-946), from the authors' preprint dated 12 May 2006; worth reading as the interventional counterpart of transfer entropy: information flow I(X_A → X_B | X̂_S) is conditional mutual information computed from do()-distributions instead of observational ones, it vanishes exactly under a graphical condition (ud-separation), and examples show it differs from transfer entropy and conditional mutual information in both directions. Read it knowing that it needs the full causal Bayesian network and its mechanisms, not data; that the examples are deterministic binary networks; and that its Example 6 claim of one bit holds, by the record's calculation, only for k = 1 or when imposing X_k alone. The published version (Advances in Complex Systems 11(1), 2008) was not compared. Filed as THEORY-710.
Tagscausality, information-theory
Cited as “paper” byNOTE-946 — Ay and Polani — Information flows in causal networks
Cited as “source” byTHEORY-710 — In a finite causal Bayesian network, a set of nodes is causally independent of another, imposing a third, for every choice of mechanisms exactly when every directed path between them passes through the third; information flow, conditional mutual information computed from interventional instead of observational distributions, vanishes under causal independence, and it can differ from transfer entropy and conditional mutual information in either direction

Graph View

Nihat Ay and Daniel Polani (2008), “Information Flows in Causal Networks”, Advances in Complex Systems 11(1): 17–41 — doi:10.1142/S0219525908001465

Key takeaways

From the reading of the 2006 preprint, NOTE-946.

  • Causal independence (Eq. 4). X_B ⫫ X_A | X̂_S if p(x_B | x̂_A, x̂_S) = Σ_{x′_A} p(x′_A | x̂_S) p(x_B | x̂′_A, x̂_S) for all x_A, x_S. Defined through mechanisms, so for every configuration, not only observed ones. Not symmetric.
  • Theorem 2. B is ud-separated from A by S (every directed path from A to B meets S) iff X_B is causally independent of X_A imposing X_S for every G-causal model. The interventional analogue of Verma and Pearl’s d-separation theorem. d-separation implies ud-separation (Prop. 1).
  • Information flow (§5). I(X_A → X_B | X̂_S) is the observational conditional mutual information’s formula with each conditional replaced by its interventional counterpart, averaged over the observational p(x_S). Zero under causal independence; the converse needs every x_S to have positive probability (Prop. 3).
  • It is not transfer entropy (Example 6). Two binary chains that swap states each step have zero transfer entropy and one bit of flow (at k = 1; see below). Two independent inverters started from the same distribution are observationally identical and have zero of both. Imposing a node can raise flow (Example 3) or lower it (Example 4).
  • Markov chains (§6). For a single chain, flow between time windows equals their mutual information.

Standing in the record

Filed on 2026-10-17 and read the same day. Filed as THEORY-710. It is the information-theoretic form of the interventional notion Eichler and Didelez use (THEORY-704). The reading’s check of Example 6 found the paper’s one-bit flow holds for k = 1, and for any k only when the target’s last value alone is imposed. Imposing the whole past X_[1,k] fixes Y_k = x_{k−1} and gives zero for k ≥ 2 by the paper’s own definition. That is the record’s calculation, not the authors’.

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