Missing data is almost always present in real datasets, and introduces several statistical issues. One fundamental issue is that, in the absence of strong uncheckable assumptions, effects of interest are typically not nonparametrically identified. In this article, we review the generic approach of the use of identifying restrictions from a likelihood-based perspective, and provide points of contact for several recently proposed methods. An emphasis of this review is on restrictions for nonmonotone missingness, a subject that has been treated sparingly in the literature. We also present a general, fully-Bayesian, approach which is widely applicable and capable of handling a variety of identifying restrictions in a uniform manner.
Keywords: MNAR; missing data; mixture models; multiple imputation; non-ignorable missingness; nonparametric Bayes.