Differential cumulants, hierachical models and monomial ideals

TL;DR

This paper explores the algebraic structure of hierarchical models through differential cumulants and monomial ideals, revealing a deep connection between statistical independence conditions and algebraic geometry.

Contribution

It establishes an isomorphism between hierarchical models and monomial ideals using algebraic differential duality, linking statistical independence to algebraic structures.

Findings

Hierarchical models correspond to specific monomial ideals.

Differential cumulants encode independence and conditional independence.

Algebraic duality maps statistical conditions to ideal conditions.

Abstract

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.