Log In Sign Up

Hierarchical Aitchison-Silvey models for incomplete binary sample spaces

by   Anna Klimova, et al.

Multivariate sample spaces may be incomplete Cartesian products, when certain combinations of the categories of the variables are not possible. Traditional log-linear models, which generalize independence and conditional independence, do not apply in such cases, as they may associate positive probabilities with the non-existing cells. To describe the association structure in incomplete sample spaces, this paper develops a class of hierarchical multiplicative models which are defined by setting certain non-homogeneous generalized odds ratios equal to one and are named after Aitchison and Silvey who were among the first to consider such ratios. These models are curved exponential families that do not contain an overall effect and, from an algebraic perspective, are non-homogeneous toric ideals. The relationship of this model class with log-linear models and quasi log-linear models is studied in detail in terms of both statistics and algebraic geometry. The existence of maximum likelihood estimates and their properties, as well as the relevant algorithms are also discussed.


page 1

page 2

page 3

page 4


On the maximum likelihood estimation in general log-linear models

General log-linear models specified by non-negative integer design matri...

BELIEF in Dependence: Leveraging Atomic Linearity in Data Bits for Rethinking Generalized Linear Models

Two linearly uncorrelated binary variables must be also independent beca...

Generalized Cut Polytopes for Binary Hierarchical Models

Marginal polytopes are important geometric objects that arise in statist...

Quasi-independence models with rational maximum likelihood estimates

We classify the two-way independence quasi-independence models (or indep...

Logarithmic Voronoi cells

We study Voronoi cells in the statistical setting by considering preimag...

Log-linear models independence structure comparison

Log-linear models are a family of probability distributions which captur...

Exponential Hilbert series and hierarchical log-linear models

Consider a hierarchical log-linear model, given by a simplicial complex,...