  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.
