Granger causality and transfer entropy are equivalent for Gaussian variables.

By Lionel C. Barnett, Adam B. Barrett, Anil K. Seth. Physical Review Letters, 2009

1,071 citations recorded by the source corpus.

Granger causality is a statistical notion of causal influence based on prediction via vector autoregression. Developed originally in the field of econometrics, it has since found application in a broader arena, particularly in neuroscience. More recently transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes, has gained traction in a similarly wide field. While it has been recognized that the two concepts must be related, the exact relationship has until now not been formally described. Here we show that for Gaussian variables, Granger causality and transfer entropy are entirely equivalent, thus bridging autoregressive and information-theoretic approaches to data-driven causal inference.

Subfields: Cognitive Neuroscience, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d4eb18d616b212639926aa8c82e512aa91bd5606

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Citations in the corpus

Phase transfer entropy: A novel phase-based measure for directed connectivity in networks coupled by oscillatory interactions (2014, background): “Despite these differences, it has been argued that TE and GC measure the same underlying quantity (Barnett et al., 2009; Seghouane and Amari, 2012).”
Equivalence of Granger Causalityand Transfer Entropy: A Generalization. (2011, methodology): “Proof: The proof uses some ideas of the proof from [3] for Gaussian distributions.”
A MATLAB toolbox for Granger causal connectivity analysis. (2010, neutral): “One implication of this result is that, under Gaussian assumptions, there is nothing extra to account for by nonlinear extensions to G-causality, since a stationary Gaussian autoregressive process is necessarily linear (Barnett et al., 2009).”
Application of Graph Theory for Identifying Connectivity Patterns in Human Brain Networks: A Systematic Review (2019, background): “Although TE and GC are relatively equivalent for Gaussian variables (Barnett and Seth, 2009), TE needs much less computational time than GC for high model orders and greater numbers of nodes.”
Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems (2016, methodology): “In the special case of Gaussian dynamical variables, the two methods, one of the autoregressive nature (Granger test) and another based on information theoretic concepts (transfer entropy, to be discussed below), are in fact equivalent to each other [317].”
Wiener-Granger Causality: A well established methodology (2011, positive): “It has recently been shown that, for Gaussian variables, WGC and TE are in fact entirely equivalent (Barnett et al., 2009).”
p-Causality: Identifying Spatiotemporal Causal Pathways for Air Pollutants with Urban Big Data (2016, methodology): “For variables obeying Gaussian distribution, Granger causality is in the same form as conditional mutual information [2], which has been used successfully for constructing structures for Bayesian networks.”
A statistical framework for neuroimaging data analysis based on mutual information estimated via a gaussian copula (2016, background): “This is the information theoretic analogue of Granger causality [Barnett et al., 2009; Granger, 1969] and following the arguments developed by Wiener and Granger [Granger, 1969; Wiener, 1956] can be used to infer causal relationships between brain signals, with some caveats [Bressler and Seth, 2011;”
A study of problems encountered in Granger causality analysis from a neuroscience perspective (2017, methodology): “These approaches are strictly time-domain measures, and their similarity to GG causality (31) suggests that they would face similar problems with interpretation such as independence of receiver dynamics.”
Granger Causality Analysis in Neuroscience and Neuroimaging (2015, background): “pendency or shared information between two variables (Barnett et al., 2009).”
Analysing connectivity with Granger causality and dynamic causal modelling (2013, methodology): “GC applied to local-field potentials (LFPs) recorded from cat visual cortex; important early clarification of statistical issues [23] GC analysis of LFP data obtained from awake monkeys reveal directional beta-frequency interactions in a large –scale network during motor maintenance behaviour [25] N”
Causal network reconstruction from time series: From theoretical assumptions to practical estimation. (2018, background): “A more general definition is that of (bivariate) transfer entropy (Schreiber, 2000; Barnett et al., 2009) ITEbivX→Y = I(X−t ; Yt | Y−t ), (2) where I(X ; Y | Z) denotes the conditional mutual information (CMI).”
Analysis of sampling artifacts on the Granger causality analysis for topology extraction of neuronal dynamics (2014, background): “When time series are linear and Gaussian, GC is equivalent to the transfer entropy (Barnett et al., 2009; Bressler and Seth, 2011).”
JIDT: An Information-Theoretic Toolkit for Studying the Dynamics of Complex Systems (2014, positive): “For multivariate Gaussians, the TE is equivalent (up to a factor of 2) to the Granger causality (Barnett et al., 2009).”