Transportability from Multiple Environments with Limited Experiments: Completeness Results

This paper addresses the problem of $mz$-transportability, that is, transferring causal knowledge collected in several heterogeneous domains to a target domain in which only passive observations and limited experimental data can be collected. The paper first establishes a necessary and sufficient condition for deciding the feasibility of $mz$-transportability, i.e., whether causal effects in the target domain are estimable from the information available. It further proves that a previously established algorithm for computing transport formula is in fact complete, that is, failure of the algorithm implies non-existence of a transport formula. Finally, the paper shows that the do-calculus is complete for the $mz$-transportability class.