Random variables with completely independent subcollections

We investigate the algebra and geometry of the independence conditions on discrete random variables in which we consider a collection of random variables and study the condition of independence of some subcollections. We interpret independence conditions as an ideal of algebraic relations. After a change of variables, this ideal is generated by generalized 2×2 minors of multi-way tables and linear forms. In particular, let Δ be a simplicial complex on some random variables and A be the table corresponding to the product of those random variables. If A is Δ-independent table then A can be written as the entrywise sum AI+A0 where AI is a completely independent table and A0 is identically 0 in its Δ-margins. We compute the isolated components of the original ideal, showing that there is only one component that could correspond to probability distributions, and relate the algebra and geometry of the main component to that of the Segre embedding. If Δ has fewer than three facets, we are able to compute generators for the main component, show that it is Cohen–Macaulay, and give a full primary decomposition of the original ideal.