Source
Ay & Polani (2008), “Information Flows in Causal Networks”, Advances in Complex Systems 11(1): 17–41, LIT-1174, read in full in NOTE-946 from the authors’ preprint of 12 May 2006. The journal version was not compared.
What was actually shown
Setting. A DAG with finite state sets and known mechanisms p_v(x_v | x_pa(v)). Interventions by truncated factorization.
Causal independence (Eq. 4) and Theorem 2. X_B ⫫ X_A | X̂_S if setting x_A after x_S leaves X_B’s post-interventional distribution unchanged. For every model on the DAG, this holds iff B is ud-separated from A by S: every directed path from A to B meets S. d-separation implies ud-separation (Prop. 1).
Information flow (§5, Prop. 3). I(X_A → X_B | X̂_S) is the conditional mutual information formula with interventional conditionals, averaged over the observational p(x_S). It is zero under causal independence, and only then if every x_S has positive probability. Positive flow implies an unblocked directed path (Corollary 2).
Divergence from correlational measures (Examples 3–6).
- Diamond, Z = X ⊕ Y: I(X:Y) = 1 but I(X → Y) = 0; I(W:Z | Y) = 0 but I(W → Z | Ŷ) = 1.
- Same joint distribution, two mechanisms: I(X → Z | Ŷ) = 1 or ≈ 0.311, while I(X:Z | Y) = 0 in both.
- Two chains copying each other against two independent inverters, observationally identical: transfer entropy zero in both, flow one bit against zero at k = 1.
The reading recomputed every value. For Example 6 at k ≥ 2, imposing the whole past X_[1,k] gives zero flow in the copying case too, because it fixes Y_k = x_{k−1}. Imposing X_k alone restores one bit at every k. The paper does not state k.
What could have come out otherwise: Theorem 2’s “if” direction could have failed for some DAG, with an unblocked directed path but no model showing dependence. The copying construction shows one always exists.
What this does not say
- How to compute it from data. It needs the mechanisms. Identification from observational data is THEORY-704’s kind of result, not this.
- That transfer entropy is always wrong about flow. The two formulas coincide term by term when conditioning on the pasts equals setting them, for every value, and the observational distribution has full support. That is the record’s reading of the definitions (NOTE-946).
- Anything outside small deterministic examples. Determinism is where observational measures are degenerate.
- That flow is additive like matter flow. Imposing a node can raise it (diamond) or lower it (channel splitting).
Connections
- THEORY-704: the time-series identification of interventional effects. Together they say what an interventional information flow is and when observational data can deliver one. My connection.
- CLAIM-333 and Example 6 show the same gap from opposite sides: zero transfer entropy with a real effect.
- THEORY-711 (directed information) and THEORY-641 (transfer entropy as a likelihood ratio) are observational measures.
- TERM-058, TERM-057.