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Age of Incorrect Information for Remote Estimation of a Binary Markov Source

For monitoring applications, the Age of Information (AoI) metric has been the primary focus of recent research, but closely related to monitoring is the problem of real-time or remote estimation. Age of Information has been shown to be insufficient for minimizing remote estimation error, but recently a metric known as Age of Incorrect Information (AoII) was proposed that characterizes the cost of a monitor being in an erroneous state over time. In this work, we study the AoII metric in the simple context of monitoring a symmetric binary information source over a delay system with feedback. We compare three different performance metrics: real-time error, AoI, and AoII. For each metric, we formulate the optimal sampling problem as a Markov decision process and apply a dynamic programming algorithm to compute the optimal performance and policy. We also simulate the system for two sampling policies: sample-at-change and zero-wait, and we observe which policy coincides with the optimal policy for each metric. For a variety of delay distributions and AoII penalty functions, we observe that the optimal policy for the real-time error and for AoII are equal to the sample-at-change policy, whereas the optimal policy for AoI is a threshold policy.