and the conventions p ( y 0 | y 0 , ⋯ ⋯ , y − − 1 ) = p ( y 0 ) and p ( x 0 | y 0 , ⋯ ⋯ , y − − 1 ) = p ( x 0 ) , for k = 0. Replacing p ( x k | y 0 , ⋯ ⋯ , y k − − 1 ) d x k by the empirical approximation