We consider the problem of identifying the causal direction between two discrete random variables using observational data. Unlike previous work, we keep the most general functional model but make an assumption on the unobserved exogenous variable: Inspired by Occam’s razor, we assume that the exogenous variable is simple in the true causal direction. We quantify simplicity using Rényi entropy. Our main result is that, under natural assumptions, if the exogenous variable has low H0 entropy (cardinality) in the true direction, it must have high H0 entropy in the wrong direction. We establish algorithmic hardness results about estimating the minimum entropy exogenous variable.We show that the problem of finding the exogenous variable with minimum entropy is equivalent to the problem of finding minimum joint entropy given n marginal distributions, also known as minimum entropy coupling
problem. We propose an efficient greedy algorithm for the minimum entropy coupling problem that provably finds a local optimum. This gives a greedy algorithm for
finding the exogenous variable with minimum H1 (Shannon
Entropy). Our greedy entropy-based causal inference algorithm
has similar performance to the state of the art additive
noise models in real datasets. One advantage of our approach
is that we make no use of the values of random variables but
only their distributions. Our method can therefore be used for
causal inference for both ordinal and also categorical data,
unlike additive noise models.