  The age of information, as a metric for evaluating information freshness, has
received a lot of attention. Recently, an interesting connection between the
age of information and remote estimation error was found in a sampling problem
of Wiener processes: If the sampler has no knowledge of the signal being
sampled, the optimal sampling strategy is to minimize the age of information;
however, by exploiting causal knowledge of the signal values, it is possible to
achieve a smaller estimation error. In this paper, we extend a previous study
by investigating a problem of sampling a stationary Gauss-Markov process,
namely the Ornstein-Uhlenbeck (OU) process. The optimal sampling problem is
formulated as a constrained continuous-time Markov decision process (MDP) with
an uncountable state space. We provide an exact solution to this MDP: The
optimal sampling policy is a threshold policy on instantaneous estimation error
and the threshold is found. Further, if the sampler has no knowledge of the OU
process, the optimal sampling problem reduces to an MDP for minimizing a
nonlinear age of information metric. The age-optimal sampling policy is a
threshold policy on expected estimation error and the threshold is found. These
results hold for (i) general service time distributions of the queueing server
and (ii) sampling problems both with and without a sampling rate constraint.
Numerical results are provided to compare different sampling policies.
