  We investigate methods for parameter learning from incomplete data that is
not missing at random. Likelihood-based methods then require the optimization
of a profile likelihood that takes all possible missingness mechanisms into
account. Optimzing this profile likelihood poses two main difficulties:
multiple (local) maxima, and its very high-dimensional parameter space. In this
paper a new method is presented for optimizing the profile likelihood that
addresses the second difficulty: in the proposed AI&M (adjusting imputation and
mazimization) procedure the optimization is performed by operations in the
space of data completions, rather than directly in the parameter space of the
profile likelihood. We apply the AI&M method to learning parameters for
Bayesian networks. The method is compared against conservative inference, which
takes into account each possible data completion, and against EM. The results
indicate that likelihood-based inference is still feasible in the case of
unknown missingness mechanisms, and that conservative inference is
unnecessarily weak. On the other hand, our results also provide evidence that
the EM algorithm is still quite effective when the data is not missing at
random.
