Note that n i=1 E θ (f A→α |x i ) is a quantity calculated in the Inside-Outside algorithm (Lari and Young, 1990) and P( y| x) is easily computed as a by-product of the same dynamic programming calculation.
, (y n , x n )), the maximum (joint) likelihood estimate (MLE) of θ is: (1) it turns out there is another maximum likelihood estimation method which maximizes the conditional likelihood or "pseudo-likelihood" of the training data (Besag, 1975).