  We present a sample path dependent measure of causal influence between time
series. The proposed causal measure is a random sequence, a realization of
which enables identification of specific patterns that give rise to high levels
of causal influence. We show that these patterns cannot be identified by
existing measures such as directed information (DI). We demonstrate how
sequential prediction theory may be leveraged to estimate the proposed causal
measure and introduce a notion of regret for assessing the performance of such
estimators. We prove a finite sample bound on this regret that is determined by
the worst case regret of the sequential predictors used in the estimator.
Justification for the proposed measure is provided through a series of
examples, simulations, and application to stock market data. Within the context
of estimating DI, we show that, because joint Markovicity of a pair of
processes does not imply the marginal Markovicity of individual processes,
commonly used plug-in estimators of DI will be biased for a large subset of
jointly Markov processes. We introduce a notion of DI with "stale history",
which can be combined with a plug-in estimator to upper and lower bound the DI
when marginal Markovicity does not hold.
