Appears in collection : 2016 - T1 - WS4 - Inference problems theme

The problem of estimating the directed information rate between two Markov chains of arbitrary (but finite) order is considered. Specifically for the so-called “plug-in” (or maximum-likelihood) estimator, under natural conditions we show that it is consistent with probability one, and that it is asymptotically Gaussian. From this it is show that its convergence rate is of O(1/n√), which is the best possible. A connection is established between this estimation problem and that of performing a hypothesis test for the presence of causal influence between the two processes. Under the null hypothesis, which corresponds to the absence of (temporal) causality, we show that the plug-in estimator has an asymptotic χ2distribution, and that this estimator can be expressed precisely in terms of the classical likelihood ratio statistic. Combining these two results facilitates the design of a Neyman-Pearson likelihood ratio test for the presence of causal influence. This is joint work with Maria Skoularidou.