  This paper proposes using the uncertainty of information (UoI), measured by
Shannon's entropy, as a metric for information freshness. We consider a system
in which a central monitor observes multiple binary Markov processes through a
communication channel. The UoI of a Markov process corresponds to the monitor's
uncertainty about its state. At each time step, only one Markov process can be
selected to update its state to the monitor; hence there is a tradeoff among
the UoIs of the processes that depend on the scheduling policy used to select
the process to be updated. The age of information (AoI) of a process
corresponds to the time since its last update. In general, the associated UoI
can be a non-increasing function, or even an oscillating function, of its AoI,
making the scheduling problem particularly challenging. This paper investigates
scheduling policies that aim to minimize the average sum-UoI of the processes
over the infinite time horizon. We formulate the problem as a restless
multi-armed bandit (RMAB) problem, and develop a Whittle index policy that is
near-optimal for the RMAB after proving its indexability. We further provide an
iterative algorithm to compute the Whittle index for the practical deployment
of the policy. Although this paper focuses on UoI scheduling, our results apply
to a general class of RMABs for which the UoI scheduling problem is a special
case. Specifically, this paper's Whittle index policy is valid for any RMAB in
which the bandits are binary Markov processes and the penalty is a concave
function of the belief state of the Markov process. Numerical results
demonstrate the excellent performance of the Whittle index policy for this
class of RMABs.
