Optimality, accuracy, and efficiency of an exact functional test

Functional dependency can lead to discoveries of new mechanisms not possible via symmetric association. Most asymmetric methods for causal direction inference are not driven by the function-versus-independence question. A recent exact functional test (EFT) was designed to detect functionally dependent patterns model-free with an exact null distribution. However, the EFT lacked a theoretical justification, had not been compared with other asymmetric methods, and was practically slow. Here, we prove the functional optimality of the EFT statistic, demonstrate its advantage in functional inference accuracy over five other methods, and develop a branch-and-bound algorithm with dynamic and quadratic programming to run at orders of magnitude faster than its previous implementation. Our results make it practical to answer the exact functional dependency question arising from discovery-driven artificial intelligence applications. Software that implements EFT is freely available in the R package 'FunChisq' (=2.5.0) at https://cran.r-project.org/package=FunChisq.