A forward-backward probability hypothesis density (PHD) smoother involving forward filtering followed by backward smoothing is proposed. The forward filtering is performed by Mahler's PHD recursion. The PHD backward smoothing recursion is derived using finite set statistics (FISST) and standard point process theory. Unlike the forward PHD recursion, the proposed backward PHD recursion is exact and does not require the previous iterate to be Poisson. In addition, assuming the previous iterate is Poisson, the cardinality distribution and all moments of the backward-smoothed multi-target density are derived. It is also shown that PHD smoothing alone does not necessarily improve cardinality estimation. Using an appropriate particle implementation we present a number of experiments to investigate the ability of the proposed multi-target smoother to correct state as well as cardinality errors.
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