We are seeking the maximum likelihood estimate of each ϕ, given fixed alignments π: To find this maximizer for any given π , we need to find a marginal distribution over the edges connecting any two languages a and d. With this distribution, we calculate the expected "alignment unigrams."
for each pair of phonemes x and y (or empty phoneme ε), we need to find the quantity: where we denote #(x, y; z) to be the number of times the pair of phonemes (x, y) are aligned in alignment z.