Is the direction of greater Granger causal influence the same as the direction of information flow?

Granger causality is an established statistical measure of the “causal influence” that one stochastic process X has on another process Y. Along with its more recent generalization - Directed Information - Granger Causality has been used extensively in neuroscience, and in complex interconnected systems in general, to infer statistical causal influences. More recently, many works compare the Granger causality metrics along forward and reverse links (from X to Y and from Y to X), and interpret the direction of greater causal influence as the “direction of information flow”. In this paper, we question whether the direction yielded by comparing Granger Causality or Directed Information along forward and reverse links is always the same as the direction of information flow. We explore this question using two simple theoretical experiments, in which the true direction of information flow (the “ground truth”) is known by design. The experiments are based on a communication system with a feedback channel, and employ a strategy inspired by the work of Schalkwijk and Kailath. We show that in these experiments, the direction of information flow can be opposite to the direction of greater Granger causal influence or Directed Information. We also provide information-theoretic intuition for why such counterexamples are not surprising, and why Granger causality-based information-flow inferences will only get more tenuous in larger networks. We conclude that one must not use comparison/difference of Granger causality to infer the direction of information flow.