Differential cumulants, hierarchical models and monomial ideals
Daniel Bruynooghe · London School of Economics and Political Science Theses Online (London School of Economics and Political Science) · 2011
This thesis studies hierarchical models for the joint density f X of a random vector X = (X 1 , ..., X d ), i.e. models characterised by the fact that interaction within a set of variables implies interaction within any of its subsets.A hierarchical model on a simplicial complex S can be written in exponential form asThe statistical implications of the choice of S are studied.Associated to a simplicial complex S is its Stanley-Reisner ideal I S .Hence, hierarchical models can be identied uniquely with monomial ideals.This isomorphism bridges the elds of statistics and commutative algebra.Simplicial complexes holding sets with at most two elements can be illustrated with graphs, thus leading to graphical models.The missing edges of a graph represent two-element sets excluded from S implying conditional independence.It is shown that sets excluded from S imply dierential conditions on the log-density and that these dierentials arise naturally as cumulants in an innitesimally small neighbourhood around a given x 0 ∈ R d .A new bootstrap test for conditional independence is constructed based on the notion that certain dierential cumulants are zero everywhere under conditional independence.To Vlad, who is sadly missed.To plagiarise Michael Atiyah, Henry peered around a lot of corners.He also taught me a lot about life in general.Henry's dedication and commitment was outstanding.I could not have hoped for better supervision.I would like to thank Lenny Smith for his continued support throughout the project.He taught