Causal discovery in the presence of missing data

Missing data are ubiquitous in many domains such as healthcare. When these data entries are not missing completely at random, the (conditional) independence relations in the observed data may be different from those in the complete data generated by the underlying causal process. As a consequence, simply applying existing causal discovery methods to the observed data may lead to the wrong conclusions. In this paper, we aim at developing a causal discovery method to recover the underlying causal structure from data sets with values that are missing with different mechanisms, including missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). With missingness mechanisms represented by a missingness graph (m-graph), we analyze conditions under which additional correction is needed to derive conditional independence/dependence relations in the complete data. Based on our analysis, we propose Missing Value PC (MVPC), which incorporates additional corrections with the PC algorithm, a frequently-used causal discovery method. Our proposed MVPC is shown in theory to give asymptotically correct results even on data that are MAR or MNAR. Experimental results on both synthetic data and real healthcare applications illustrate that the proposed algorithm is able to find correct causal relations even in the general case of MNAR.