  In many practical studies, learning directionality between a pair of
variables is of great interest while notoriously hard when their underlying
relation is nonlinear. This paper presents a method that examines asymmetry in
exposure-outcome pairs when a priori assumptions about their relative ordering
are unavailable. Our approach utilizes a framework of generative exposure
mapping (GEM) to study asymmetric relations in continuous exposure-outcome
pairs, through which we can capture distributional asymmetries with no prefixed
variable ordering. We propose a coefficient of asymmetry to quantify relational
asymmetry using Shannon's entropy analytics as well as statistical estimation
and inference for such an estimand of directionality. Large-sample theoretical
guarantees are established for cross-fitting inference techniques. The proposed
methodology is extended to allow both measured confounders and contamination in
outcome measurements, which is extensively evaluated through extensive
simulation studies and real data applications.
