  We present a sample path dependent measure of causal influence between two
time series. The proposed measure is a random variable whose expected sum is
the directed information. A realization of the proposed measure may be used to
identify the specific patterns in the data that yield a greater flow of
information from one process to another, even in stationary processes. We
demonstrate how sequential prediction theory may be leveraged to obtain
accurate estimates of the causal measure at each point in time and introduce a
notion of regret for assessing the performance of estimators of the measure. We
prove a finite sample bound on this regret that is determined by the regret of
the sequential predictors used in obtaining the estimate. We estimate the
causal measure for a simulated collection of binary Markov processes using a
Bayesian updating approach. Finally, given that the measure is a function of
time, we demonstrate how estimators of the causal measure may be extended to
effectively capture causality in time-varying scenarios.
