Differential cumulants, hierachical models and monomial ideals

For a joint probability density function f(x) of a random vector X the mixed partial derivatives of log f(x) can be interpreted as limiting cumulants in an infinitesimally small open neighborhood around x. Moreover, setting them to zero everywhere gives independence and conditional independence conditions. The latter conditions can be mapped, using an algebraic differential duality, into monomial ideal conditions. This provides an isomorphism between hierarchical models and monomial ideals. It is thus shown that certain monomial ideals are associated with particular classes of hierarchical models.