Recovering Probability Distributions from Missing Data

A probabilistic query may not be estimable from observed data corrupted by missing values if the data are not missing at random (MAR). It is therefore of theoretical interest and practical importance to determine in principle whether a probabilistic query is estimable from missing data or not when the data are not MAR. We present algorithms that systematically determine whether the joint probability distribution or a target marginal distribution is estimable from observed data with missing values, assuming that the data-generation model is represented as a Bayesian network, known as m-graphs, that not only encodes the dependencies among the variables but also explicitly portrays the mechanisms responsible for the missingness process. The results significantly advance the existing work.