  In this paper, we consider a discrete memoryless point to point channel with
noisy feedback, where there is a sender with a private message that she wants
to communicate to a receiver by sequentially transmitting symbols over a noisy
channel. After each transmission, she receives a noisy feedback of the symbol
received by the receiver. The goal is to design transmission control strategy
of the sender that minimize the average probability of error. This is an
instance of decentralized control of information where the two controllers, the
sender and the receiver have no common information. There exist no methodology
in the literature that provides a notion of "state" and a dynamic program to
find optimal policies for this problem In this paper, we show introduce a
notion of state, based on which we provide a sequential decomposition
methodology that finds optimum policies within the class of Markov strategies
with respect to this state (which need not be globally optimum). This allows to
decompose the problem across time and reduce the complexity dependence on time
from double exponential to linear in time.
