  We consider the problem of transmission scheduling for the remote estimation
of a discrete-time autoregressive Markov process that is driven by white
Gaussian noise. A sensor observes this process, and then decides to either
encode the current state of this process into a data packet and attempts to
transmit it to the estimator over an unreliable wireless channel modeled as a
Gilbert-Elliott channel, or does not send any update. Each transmission attempt
consumes $\lambda$ units of transmission power, and the remote estimator is
assumed to be linear. The channel state is revealed only via the feedback
(ACK\slash NACK) of a transmission, and hence the channel state is not revealed
if no transmission occurs. The goal of the scheduler is to minimize the
expected value of an infinite-horizon cumulative discounted cost, in which the
instantaneous cost is composed of the following two quantities: (i)~squared
estimation error, (ii) transmission power. We show that this problem can
equivalently be posed as a partially observable Markov decision process
(POMDP), in which the scheduler maintains a belief about the current state of
the channel, and makes decisions on the basis of the current value of the
estimation error, and the belief state.~We then show that the optimal policy is
of threshold-type, i.e. for each value of the estimation error $e$, there is a
threshold $b\ust(e)$ such that when the error is equal to $e$, then it is
optimal to transmit only when the current belief state is greater than
$b\ust(e)$.
