In other words, given ξ ξ n ( N ) the samples ξ ξ n + 1 ( N ) are independent random variables with probability distribution Φ Φ ( η η n N ) . The rationale behind this mean field simulation technique is the following: We expect that when η η n N is a good approximation of η η n , then Φ Φ ( η η n N ) is an approximation of Φ Φ ( η η n ) = η η n + 1 . Thus, since η η n + 1 N is the empirical measure of N conditionally independent random variables with common probability distribution Φ Φ ( η η n N ) , we expect η η n + 1 N to be a good approximation of η η n + 1 .