In the above displayed formula, p ^ ^ ( d x k − − 1 | ξ ξ k i , ( y 0 , ⋯ ⋯ , y k − − 1 ) ) stands for the conditional distribution p ^ ^ ( d x k − − 1 | x k , ( y 0 , ⋯ ⋯ , y k − − 1 ) ) evaluated at x k = ξ ξ k i . In the same vein, p ( y k − − 1 | ξ ξ k − − 1 j ) and p ( ξ ξ k i | ξ ξ k − − 1 j ) stand for the conditional densities p ( y k − − 1 | x k − − 1 ) and p ( x k | x k − − 1 ) evaluated at x k = ξ ξ k i and x k − − 1 = ξ ξ k − − 1 j . These models allows to reduce integration with respect to the densities p ( ( x 0 , ⋯ ⋯ , x n ) | ( y 0 , ⋯ ⋯ , y n − − 1 ) ) in terms of matrix operations with respect to the Markov transitions of the chain described above. For instance, for any function f k we have the particle estimates