The forward-backward Probability Hypothesis Density smoother

A forward-backward Probability Hypothesis Density (PHD) smoother involving forward filtering followed by backward smoothing is derived. 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.