The so-called complete version of the log-likelihood function (10) assumes that the hidden (missing) alignments are also known:
The EM algorithm maximises (10) iteratively, through the application of two basic steps in each iteration: the E(xpectation) step and the M(aximisation) step. At iteration , the E step computes the expected value of (12) given the observed (incomplete) data, , and a current estimation of the parameters, . This reduces to the computation of the expected value of :
Then, the M step finds a new estimate of , , by maximising (12), using (13) instead of the missing . This results in:
for all and ; where is the Kronecker delta function; i.e., if ; 0 otherwise.
An initial estimate for , , is required for the EM algorithm to start. This can be done by assuming that the translation probabilities are uniformly distributed; i.e.,