One can estimate θ by maximizing the loglikelihood of the data If the x i 's are missing, EM tries to iteratively maximize the marginal probability: The E-step of each iteration comprises computing , the expected loglikelihood of the complete data, where p θt (x | y i ) is the conditional part of p θt (x, y i ) under the current iterate θ t , and the M-step comprises maximizing it: Notice that if we replace p θt (x|y i ) with p φ (x | y i ) in the equation above, and admit negated loglikelihood as a loss function, then the EM update (14) becomes identical to (4).
the minimum imputed risk approach of Section 3.1 differs from EM in (i) using an externally-provided and static p φ , instead of refining it at each iteration based on the current p θt , and (ii) using a specific loss function, namely negated log-likelihood.