Granger causality and transfer entropy are equivalent for Gaussian variables

Lionel Barnett, Adam B. Barrett, and Anil K. Seth
Physical Review Letters (in press)

'Cause' and 'effect' are fundamental to the way we understand the world. Yet identifying causal interactions in systems, by making measurements, has proven challenging and often controversial. In this article, we show that two previously distinct approaches, 'Granger causality' and 'transfer entropy', are entirely equivalent under quite general conditions. The basic idea of Granger causality is that one variable causes another to the extent that it helps predict the other. Transfer entropy, by contrast, is based on 'resolution of uncertainty' rather than prediction. More importantly, the two concepts are implemented in very different mathematical frameworks, autoregressive modeling and information theory, which make different assumptions and offer different advantages. By showing that Granger causality and transfer entropy are equivalent, our work allows insights and techniques from one framework to be translated directly into the other. We also present a natural way to describe causal interactions between sets of variables, rather than - as is typical - between individual variables. More generally, our results strengthen the mathematical foundations underpinning causal analysis, which will help scientists expose causal interactions in complex systems of all kinds.