Note: The closest thing I can think of having seen are treatments of random variables (with finite support) as standard simplices, since each point of the simplex corresponds to a probability distribution such that $p_1 + \dots + p_n = 1$. Also, this. However, these are not what I am looking for: in what I am describing, only the vertices represent random variables -- the higher-order faces do not represent random variables, they represent relationships of statistical independence between random variables (e.g. a $2$-face connecting three vertices corresponds to the statement that the three random variables symbolized by the vertices are mutually statistically independent).
This question on Math.SE comes the closest to discussing the type of phenomenon I am talking about (the statistical independence of a family of random variables being modeled by an independence system, in this case the independence system of linearly independent vectors).
These two questions on Math.SE are also related, perhaps the first more so than than second: (1) (2). These two questions on MathOverflow also seem possibly related, although to be honest I don't feel I understand them well enough to discern that accurately: (1)(2).
Pages 4-5 here discussing log-linear models seem like they might be related, although it is hard for me to tell based on what is written and it is not expounded upon in much depth. In any case, like an aforementioned related question on Math.SE, it also suggests a relationship between this topic and Bayesian networks and/or algebraic statistics. However, it is worth noting that the edges in a Bayesian network denote conditional dependence, not independence, so they cannot be directly extended to the type of independence/abstract simplicial complex I am referring to. On the other hand, if one forms a new graph by connecting each node with the complement of its Markov blanket, then perhaps this might work. I.e. in other words Bayesian networks may encode the necessary information about conditional independence relationships for this to work -- on the other hand, it might still not work since "conditional independence" refers to a different (conditional) probability measure for each pair of nodes, while the standard example assumes a single choice of probability measure on the probability space on which all of the random variables are defined (I think). The answer is probably obvious but I need to think about it more.
Disclaimer: This is a revised version of my now-deleted week-old unanswered question on Math.SE -- perhaps this is a current topic of research, which perhaps is why it was unanswered on Math.SE and why it might be on-topic here. If you disagree, please let me know.
