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LIT-1171: Granger causality and transfer entropy are equivalent for Gaussian variables

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Properties6
StatusActive — read in full 2026-10-16 (NOTE-934); worth reading as the short, exact source of the Gaussian equivalence the record already leans on through Barnett and Bossomaier (NOTE-053): for jointly Gaussian stationary processes, Geweke's generalized-variance Granger causality is exactly twice the transfer entropy, conditional and multivariate. Read it knowing that the identity is about population quantities at fixed lags, that sample agreement depends on how the entropies are estimated, and that the remark that a stationary Gaussian AR process leaves nothing for nonlinear Granger causality cites unpublished work. Filed as THEORY-700.
Tagsinformation-theory, causality
ExtendsLIT-1173 — Investigating Causal Relations by Econometric Models and Cross-spectral Methods, LIT-1169 — Measurement of Linear Dependence and Feedback between Multiple Time Series
Extended byLIT-048 — Transfer Entropy as a Log-likelihood Ratio
Cited as “paper” byNOTE-934 — Barnett, Barrett and Seth — Granger causality and transfer entropy are equivalent for Gaussian variables
Cited as “source” byTHEORY-700 — For jointly Gaussian stationary processes, Granger causality in Geweke's generalized-variance form is exactly twice the transfer entropy, conditional and multivariate, at the regressions' lags, because least-squares residual covariance is the partial covariance and a Gaussian's conditional entropy is half its log-determinant plus a constant

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Lionel Barnett, Adam B. Barrett and Anil K. Seth (2009), “Granger causality and transfer entropy are equivalent for Gaussian variables”, Physical Review Letters 103: 238701 — ARXIV-0910.4514

Key takeaways

From the reading, NOTE-934.

  • The identity (Eq. 13). If X, Y and Z are jointly Gaussian and stationary, F_{Y→X|Z} = 2T_{Y→X|Z}, with F in Geweke’s generalized-variance form (Eq. 9) and T the conditional transfer entropy (Eq. 10). It holds for multivariate variables and reduces to the standard univariate statistic (Eq. 8).
  • Why. The OLS residual covariance is the partial covariance Σ(X|Y) = Σ(X) − Σ(X,Y)Σ(Y)⁻¹Σ(X,Y)ᵀ (Eq. 3), and a Gaussian’s conditional entropy is ½ln|Σ(X|Y)| + ½n ln(2πe) (Eq. 11), by the block determinant identity.
  • Prediction, not intervention. The authors say a G-causality interaction is not a physically instantiated causal interaction, which “can only be unambiguously identified by perturbing a system” (p. 2). The regressions are predictive models, not claims about the data-generating process, and may omit latent or exogenous variables (fn. 31).
  • In sample. Numerical agreement depends on the transfer-entropy estimator. With covariance-based entropies it is guaranteed; with partition, kernel or nearest-neighbour estimators it is not (p. 7).

Standing in the record

Filed on 2026-10-16 as the paper NOTE-053 and THEORY-641 cite for “T̂ is half the Granger statistic”, and read the same day. The reading confirms the citation. Filed as THEORY-700, which THEORY-641 generalizes beyond the Gaussian case. causality is for its subject, the Wiener–Granger notion of causal inference, which the authors themselves keep apart from intervention. It carries no instruction for machine-learning practice.

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