The terminology "mean-field approximation" comes from the fact that we replace at each time step the probability measure p ( d x k | y 0 , ⋯ ⋯ , y k − − 1 ) by the empirical approximation p ^ ^ ( d x k | y 0 , ⋯ ⋯ , y k − − 1 ) . The mean-field particle approximation of the filtering problem is far from being unique. Several strategies are developed in the books.