Filtering and Smoothing State Estimation for Flag Hidden Markov Models

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 while other states are unmeasured. For flag HMMs, an explicit computation of the probability of error for the maximum-likelihood smoother is developed. Also, some structural results are obtained for maximum likelihood detectors and their error probabilities. These algebraic and structural results are leveraged to address sensor placement in three examples, including one on activity-monitoring in a home environment that is drawn from field data.