Four estimators of the directed information rate between a pair of jointly
stationary ergodic finite-alphabet processes are proposed, based on universal
probability assignments. The first one is a Shannon--McMillan--Breiman type
estimator, similar to those used by Verd\'u (2005) and Cai, Kulkarni, and
Verd\'u (2006) for estimation of other information measures. We show the almost
sure and $L_1$ convergence properties of the estimator for any underlying
universal probability assignment. The other three estimators map universal
probability assignments to different functionals, each exhibiting relative
merits such as smoothness, nonnegativity, and boundedness. We establish the
consistency of these estimators in almost sure and $L_1$ senses, and derive
near-optimal rates of convergence in the minimax sense under mild conditions.
These estimators carry over directly to estimating other information measures
of stationary ergodic finite-alphabet processes, such as entropy rate and
mutual information rate, with near-optimal performance and provide alternatives
to classical approaches in the existing literature. Guided by these theoretical
results, the proposed estimators are implemented using the context-tree
weighting algorithm as the universal probability assignment. Experiments on
synthetic and real data are presented, demonstrating the potential of the
proposed schemes in practice and the utility of directed information estimation
in detecting and measuring causal influence and delay.

提出了四种估计器用于有限字母过程对之间的有向信息速率，基于通用概率分配。利用这些理论结果，使用上下文树加权算法作为通用概率分配实现了提出的估计器，并在合成和实际数据上进行了实验，证明了提出的方案的潜力及有向信息估计在检测和测量因果影响和延迟方面的实用性。