Abstract. How can we discover whether \(X\) causes \(Y\), or vice versa, that \(Y\) causes \(X\), when we are only given a sample over their joint distribution? How can we do this such that \(X\) and \(Y\) can be univariate, multivariate, or of different cardinalities? And, how can we do so regardless of whether \(X\) and \(Y\) are of the same, or of different data type, be it discrete, numeric, or mixed? These are exactly the questions we answer in this paper.

We take an information theoretic approach, based on the Minimum Description Length principle, from which it follows that first describing the data over extit{cause} and then that of effect given cause is shorter than the reverse direction. Simply put, if \(Y\) can be explained more succinctly by a set of classification or regression trees conditioned on \(X\), than in the opposite direction, we conclude that \(X\) causes \(Y\). Empirical evaluation on a wide range of data shows that our method, Crack, infers the correct causal direction reliably and with high accuracy on a wide range of settings, outperforming the state of the art by a wide margin.