The assumption that the initial distribution and the transitions of the Markov chain are continuous for the Lebesgue measure can be relaxed. To design a particle filter we simply need to assume that we can sample the transitions X k − − 1 → → X k of the Markov chain X k , and to compute the likelihood function x k ↦ ↦ p ( y k | x k ) (see for instance the genetic selection mutation description of the particle filter given below). The continuous assumption on the Markov transitions of X k is only used to derive in an informal (and rather abusive) way different formulae between posterior distributions using the Bayes' rule for conditional densities.