Identifying Cause and Effect on Discrete Data using Additive Noise Models

Inferring the causal structure of a set of random variables from a finite sample of the joint distribution is an important problem in science. Recently, methods using additive noise models have been suggested to approach the case of continuous variables. In many situations, however, the variables of interest are discrete or even have only finitely many states. In this work we extend the notion of additive noise models to these cases. Whenever the joint distribution $\mathbf{P}^{(X,Y)}$ admits such a model in one direction, e.g. $Y = f(X)+N$, $N \perp \!\!\! \perp X$, it does not admit the reversed model $X=g(Y)+\tilde{N}$, $\tilde{N} \perp \!\!\! \perp Y$ as long as the model is chosen in a generic way. Based on these deliberations we propose an efficient new algorithm that is able to distinguish between cause and effect for a finite sample of discrete variables. We show that this algorithm works both on synthetic and real data sets.