Algebraic constraints for linear acyclic causal models

In this thesis we study the space of second- and third-order moment tensors of random vectors which satisfy a Linear Non-Gaussian Acyclic Model (LiNGAM). In such a causal model each entry X𝒾 of the random vector X corresponds to a vertex i of a directed acyclic graph (DAG) G and can be expressed as a linear combination of its direct causes (Xj : j → 𝒾) and random noise. Often the true DAG is hidden from us, and we would like to learn it. Much work has already been done in this direction. We show that there are matrices, whose entries are second- and third-order moments of X, which drop rank only if certain visible structures can be found in the graph G. This correspondence has both statistical and combinatorial interpretations. Given enough of these matrix rank constraints, we can completely determine the graph. This characterization extends previous results proven for polytrees and generalizes the well-known local Markov property for Gaussian models.