Estimating Bounds on Causal Effects Considering Unmeasured Common Causes

Maximal ancestral graphs (MAGs) can represent causal relationships in systems that include unmeasured common direct causes. Constraint-based causal discovery methods are able to find solely the Markov Equivalence Class (MEC) of the causal structure given a set of observational data. To bound the total effect estimation between a pair of variables, when the MEC of the causal structure is known, the causal effect on each member in the MEC are computed, while keeping the minimum and maximum values as the lower and upper bounds for the total causal effect. However, when the modeling is done using MAGs, i.e., the MEC is encoded as a Partial Ancestral Graph (PAG), it is not always possible to find an adjustment set over some pairs of variables for the computation of the causal effect by covariance adjustment. In such cases, the LV-IDA algorithm returns missing values on the causal effects computation for some, and occasionally all, of the MAGs in the PAG. We present an extension of the LV-IDA algorithm, which we call the LV-IDA+ algorithm, that can compute approximated bounds of causal effects between every pair of the variables on a PAG. To achieve this, we propose a way to approximate the causal effect estimations when it is not possible to find adjustment sets for some pairs of variables on the MAGs in a PAG. We evaluate the performance of LV-IDA+ using simulated data generated by a canonical DAGs and compare with the LV-IDA algorithm. The results suggest the approximations of causal effects computed by LV-IDA+, are better than the missing values (simple NAs) returned by the LV-IDA algorithm, at least for the case of observational data generated by a canonical DAGs with latent variables.