Using θ to represent the vector of all model parameters, if our data were fully observed, we could express our learning problem as Because of the factorization of the joint distribution, this learning task would be trivial, computable in closed form from relative frequency counts.
unfortunately, in our training set the variables , o and r are hidden (not observed), leaving us with a much harder optimization problem: we want model parameters which maximize the expected likelihood of the observed data, where the expectation is taken over the hidden variables for each instance.