We address the recursive computation of the a posteriori filtering probability density function (pdf) $p_{n|n}$ in a Hidden Markov Chain (HMC) model. We first observe that the classical path $p_{n-1|n-1}$ $\rightarrow$ $p_{n|n-1}$ $\rightarrow$ $p_{n|n}$ is not the only possible one that enables to compute $p_{n|n}$ recursively, and we explore the direct, prediction-based and smoothing-based recursive loops for computing $p_{n|n}$. We next propose a common methodology for computing these equations in practice. Since each path can be decomposed into a Bayesian step and a Markovian step, in the Gaussian case these two elementary operations are implemented by Gaussian transforms, and in the general case by elementary simulation techniques. By proceeding this way we obtain in parallel, for each filtering path, one set of Kalman filter (KF) equations and one generic sequential Monte Carlo (SMC) algorithm. Finally we get four KF (two of which are original), which themselves correspond to four generic SMC algorithms (two of which are original)