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In this paper, we develop a probabilistic methodology for failure diagnosis in finite state machines based on a sequence of unreliable observations. Given prior knowledge of the input probability distribution but without actual knowledge of the applied input sequence, the core problem we consider is to choose from a pool of known, deterministic finite state machines (FSMs) the one that most likely matches the given sequence of observations. The problem becomes challenging because of sensor failures which may corrupt the observed sequence by inserting, deleting, and transposing symbols with certain probabilities (that are assumed known). We propose an efficient recursive algorithm for obtaining the most likely underlying FSM, given the possibly erroneous observed sequence. The proposed algorithm essentially allows us to perform online maximum likelihood failure diagnosis and is applicable to more general settings where one is required to choose the most likely underlying hidden Markov model (HMM) based on a sequence of observations that may get corrupted with known probabilities. The algorithm generalizes existing recursive algorithms for likelihood calculation in HMMs by allowing loops in the associated trellis diagram. We illustrate the proposed methodology using an example of diagnosis in the context of communication protocols.