Equivalence between reliable feedback communication and nonlinear filter stability

This paper further demonstrates an interplay between information theory and control theory, at the level of achievability of message-point communication schemes. We establish a relationship between reliable feedback communication and the stability of the nonlinear filter. With this, we show that a newly developed feedback communication encoder - the posterior matching scheme - achieves capacity if and only if it is reliable (i.e. any finite number of bits can be reliably decoded). By making this connection to hidden Markov models, we also provide sufficient conditions (e.g. ergodicity of the Markov process and non-degeneracy of the noisy channel) on when reliability occurs.