Concavity of mutual information rate of finite-state channels

The computation of the capacity of a finite-state channel (FSC) is a fundamental and long-standing open problem in information theory. The capacity of a memoryless channel can be effectively computed via the classical Blahut-Arimoto algorithm (BAA), which, however, does not apply to a general FSC. Recently Vontobel et al. [1] generalized the BAA to compute the capacity of a finite-state machine channel with a Markovian input. Their proof of the convergence of this algorithm, however, depends on the concavity conjecture posed in their paper. In this paper, we confirm the concavity conjecture for some special FSCs. On the other hand, we give examples to show that the conjecture is not true in general.