We extend the Blahut-Arimoto algorithm for maximizing Massey's directed
information. The algorithm can be used for estimating the capacity of channels
with delayed feedback, where the feedback is a deterministic function of the
output. In order to do so, we apply the ideas from the regular Blahut-Arimoto
algorithm, i.e., the alternating maximization procedure, onto our new problem.
We provide both upper and lower bound sequences that converge to the optimum
value. Our main insight in this paper is that in order to find the maximum of
the directed information over causal conditioning probability mass function
(PMF), one can use a backward index time maximization combined with the
alternating maximization procedure. We give a detailed description of the
algorithm, its complexity, the memory needed, and several numerical examples.

本文扩展了 Blahut-Arimoto 算法，用于最大化 Massey 的有向信息，提出了一种既适用于分析具有延迟反馈信道的能力的算法，同时也提供了上下边界序列，为了找到有向信息在因果条件概率质量函数下的最大值，我们使用了反向指数时间最大化和替代最大化过程。