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A prevalent problem in general state-space models is the approximation of thesmoothing distribution of a state, or a sequence of states, conditional on theobservations from the past, the present, and the future. The aim of this paperis to provide a rigorous foundation for the calculation, or approximation, ofsuch smoothed distributions, and to analyse in a common unifying frameworkdifferent schemes to reach this goal. Through a cohesive and generic expositionof the scientific literature we offer several novel extensions allowing toapproximate joint smoothing distribution in the most general case with a costgrowing linearly with the number of particles.

On the Forward Filtering Backward Smoothing particle approximations of the smoothing distribution in general state spaces models