All Bayesian estimates of X k follow from the posterior density p ( x k | y 0 , y 1 , . . . , y k ) . The particle filter methodology provides an approximation of these conditional probabilities using the empirical measure associated with a genetic type particle algorithm. In contrast, the Markov Chain Monte Carlo or importance sampling approach would model the full posterior p ( x 0 , x 1 , . . . , x k | y 0 , y 1 , . . . , y k ) .