Here - Tilburg University

Author and presenter
Forcina, Antonio; Dept. of Economics, Finance and Statistics, University of
Perugia, Italy
Title
Smoothness of Conditional Independence Models for Discrete Data
Abstract
The paper is about a family of conditional independence models which require
constraints on complete but non hierarchical marginal log-linear parameters. For
such models, whose dependence structure cannot be represented by any of the
known graphical separation criteria, it is not known whether the model is
smooth, so that the usual asymptotics can be applied. A model is called non
smooth when the variety which it defines in the parameter space contains points
which do not admit a local approximation by a linear space.
By exploiting results on the mixed parameterization within the exponential
family, we determine a condition which has to be satisfied for the model to be
smooth. The condition has to do with the possibility to reconstruct the joint
distribution from the set of marginal log-linear parameters in a unique way. In
technical terms, the condition require that a certain jacobian matrix has spectral
radius strictly less than 1. In the simple context when only two marginals are
involved, we show that this condition is always satisfied. In the general case, we
describe an efficient numerical test for checking whether the condition is
satisfied with high probability. This approach is illustrated with several examples
of non hierarchical conditional independence models and by a directed cyclic
graph model; we establish that all these models smooth.