State Estimation for Flag Hidden Markov Models with Imperfect Sensors

State detection is studied for a special class of flag Hidden Markov Models (HMMs), which comprise 1) an arbitrary finite-state underlying Markov chain and 2) a structured observation process wherein a subset of states emit distinct flags with some probability while other states are unmeasured. The focus of this article is to develop an explicit computation of the probability of error for the maximum-likelihood filter, specifically for the case that the sensors are imperfect. The algebraic result is leveraged to address sensor placement in a couple of examples, including one on activity-monitoring in a home environment that is drawn from field data.