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The problem of out-of-sequence measurement (OOSM) processing when the filtering technique used at the tracker is the particle filter (PF) is considered. First, an exact Bayesian algorithm for updating with OOSMs is derived. Then, the PF implementation of the exact Bayesian algorithm, called A-PF, is developed. Since A-PF is rooted in exact Bayesian inference, if the number of particles is sufficiently large, A-PF is the one (and the only one) that is able to achieve the optimal performance obtained from the in-sequence processing. This is confirmed by the simulation results. Also, it is shown that the performance of A-PF is always superior to previous (heuristic) PF-based algorithms with the same number of particles.