On the Bias of Directed Information Estimators

When estimating the directed information between two jointly stationary Markov processes, it is typically assumed that the recipient of the directed information is itself Markov of the same order as the joint process. While this assumption is often made explicit in the presentation of such estimators, a characterization of when we can expect the assumption to hold is lacking. Using the concept of d-separation from Bayesian networks, we present sufficient conditions for which this assumption holds. We further show that the set of parameters for which these conditions are not also necessary has Lebesgue measure zero. Given the strictness of these conditions, we introduce a notion of partial directed information, which can be used to bound the bias of directed information estimates when the directed information recipient is not itself Markov. Lastly we estimate this bound on simulations in a variety of settings to assess the extent to which the bias should be cause for concern.