A generalized prediction framework for granger causality

In his 1969 paper, Granger proposed a statistical definition of causality between stochastic processes. It is based on whether causal side information helps in a sequential prediction task. However, his formulation was limited to linear predictors. We describe a generalized framework, where predictions are beliefs and compare the best predictor with side information to the best predictor without side information. The difference in the prediction performance, i.e., regret of such predictors, is used as a measure of causal influence of the side information. Specifically when log loss is used to quantify each predictor's loss and an expectation over the outcomes is used to quantify the regret, we show that the directed information, an information theoretic quantity, quantifies Granger causality. We also explore a more pessimistic setup perhaps better suited for adversarial settings where minimax criterion is used to quantify the regret.