where p ( y k | ξ ξ k i ) stands for the density p ( y k | x k ) evaluated at x k = ξ ξ k i , and p ( x k + 1 | ξ ξ k i ) stands for the density p ( x k + 1 | x k ) evaluated at x k = ξ ξ k i for i = 1 , ⋯ ⋯ , N .