Estimation Entropy and Optimal Observability

We consider a pair of correlated processes {Zn}n=−∞ and {Sn}n=−∞, where the former is observable and the latter is hidden. The uncertainty in the estimation of Sn upon the finite past history of Zn−1 0 is H(Sn|Z n−1 0 ) which is a sequence of n. The limit of Cesaro mean of this sequence is called the estimation entropy. We show that the estimation entropy is the long run average entropy of the belief state on the hidden process obtained from the observation process. Estimation entropy inversely measures the observability of the hidden process through the observed process, and its minimization is the goal for optimal observability problems such as sensor scheduling. In this paper we describe such an operational characterization of estimation entropy.