Third-order moment varieties of linear non-Gaussian graphical models

Abstract In this paper, we study linear non-Gaussian graphical models from the perspective of algebraic statistics. These are acyclic causal in which each variable is a combination its direct causes and independent noise. The underlying directed graph can be identified uniquely via set second third-order moments all random vectors that lie corresponding model. Our focus on finding relations among these for given graph. We show when polytree, form toric ideal. construct explicit trek-matrices associated to 2-treks 3-treks Their entries covariances their $2$-minors define our model set-theoretically. Furthermore, prove 2-minors also generate vanishing ideal Finally, describe polytopes ideals with hidden variables.