  Causal inference using observational data is challenging, especially in the
bivariate case. Through the minimum description length principle, we link the
postulate of independence between the generating mechanisms of the cause and of
the effect given the cause to quantile regression. Based on this theory, we
develop Bivariate Quantile Causal Discovery (bQCD), a new method to distinguish
cause from effect assuming no confounding, selection bias or feedback. Because
it uses multiple quantile levels instead of the conditional mean only, bQCD is
adaptive not only to additive, but also to multiplicative or even
location-scale generating mechanisms. To illustrate the effectiveness of our
approach, we perform an extensive empirical comparison on both synthetic and
real datasets. This study shows that bQCD is robust across different
implementations of the method (i.e., the quantile regression), computationally
efficient, and compares favorably to state-of-the-art methods.
