To derive a version of this formula that is a continuous function of λ, we will need multiple translations t s,1 , ..., t s,K for each source sentence s. The general idea is to weight each of these translations by a factor w(λ, s, k), proportional to the score m λ (t s,k |s) that t s,k is assigned by the log-linear model for a given λ: where α is the smoothing factor.
in the smoothed version of the NIST function, the term I n (t s , r s ) in equation 3is replaced by k w(λ, s, k)I n (t s,k , r s ), and the term C n (t s ) is replaced by k w(λ, s, k)C n (t s,k ).