Recovering from selection bias using marginal structure in discrete models

This paper considers the problem of inferring a discrete joint distribution from a sample subject to selection, such as might arise from a case-control study. Abstractly, we want to identify a distribution p(x,w) from its condi-tional p(x |w). We introduce new assump-tions on the marginal model for p(x), un-der which generic identification is possible. These assumptions are quite general and can easily be tested; they do not require pre-cise background knowledge of p(x) or p(w), such as proportions estimated from previous studies. We particularly consider conditional independence constraints, which often arise from graphical and causal models, although other constraints can also be used. We show that generic identifiability of causal effects is possible in a much wider class of causal mod-els than had previously been known.