  In this paper we introduce two variational equalities of directed
information, which are analogous to those of mutual information employed in the
Blahut-Arimoto Algorithm (BAA). Subsequently, we introduce nonanticipative Rate
Distortion Function (RDF) ${R}^{na}_{0,n}(D)$ defined via directed information
introduced in [1], and we establish its equivalence to Gorbunov-Pinsker's
nonanticipatory $\epsilon$-entropy $R^{\varepsilon}_{0,n}(D)$. By invoking
certain results we first establish existence of the infimizing reproduction
distribution for ${R}^{na}_{0,n}(D)$, and then we give its implicit form for
the stationary case. Finally, we utilize one of the variational equalities and
the closed form expression of the optimal reproduction distribution to provide
an algorithm for the computation of ${R}^{na}_{0,n}(D)$.
