Exploring the causal order of binary variables via exponential hierarchies of Markov kernels

We propose a new algorithm for estimating the causal structure that underlies the observed dependence among n (n ≥ 4) binary variables X1, . . . , Xn. Our inference principle states that the factorization of the joint probability into conditional probabilities for Xj given X1, . . . , Xj−1 often leads to simpler terms if the order of variables is compatible with the directed acyclic graph representing the causal structure. We study joint measures of OR/AND gates and show that the complexity of the conditional probabilities (the so-called Markov kernels), defined by a hierarchy of exponential models, depends on the order of the variables. Some toy and real-data experiments support our inference rule.