We address the recursive computation of the a posteriori filtering pdf p(n|n) in a Hidden Markov Chain (HMC). Classically p(n|n) is computed via the recursion p(n-1|n-1),p(n|n-1),p(n|n). In this paper we explore direct, prediction-based (P-based) and smoothing-based (S-based) alternative loops for propagating p(n|n). We next address sequential Monte Carlo (SMC) implementations of these filtering paths, and compare our algorithms via simulations