is the probability of the random paths of a Markov chain ( X k , n ♭ ♭ ) 0 ⩽ ⩽ k ⩽ ⩽ n running backward in time from time k=n to time k=0, and evolving at each time step k in the state space associated with the population of particles ξ ξ k i , i = 1 , ⋯ ⋯ , N .