Here - Tilburg University

Author and presenter
Roverato, Alberto; Dept. of Science Statistics, University of Bologna, Italy
Title
Log-linear Moebius models for binary data
Abstract
Models of marginal independence can be useful in several contexts and
sometimes they may be used to represent independence structures induced
after marginalizing over latent variables. A relevant class of marginal models is
given by graphical models for marginal independence that use either bi-directed
or dashed undirected graphs to encode marginal independence patterns between
the variables of a random vector (Cox and Wermuth, 1993). When variables
follow a multinomial distribution, graphical models for marginal independence
are curved exponential families and the marginal independence restrictions
correspond to complicated non-linear restrictions on the parameters of the
traditional log-linear models. Parameterizations more suitable in this context
have been proposed by Drton and Richardson (2008), shortly DR2008, and by
Lupparelli, Marchetti and Bergsma (2009), shortly LMB2009. DR2008 introduced
the Moebius parameters and showed that marginal independence constraints
correspond to the factorization of certain mean parameters of the exponential
family representation of the model. Although it is not straightforward to identify
the set of factorizations corresponding to a given independence model, this
parameterization has several advantages and, in particular, the likelihood can be
written in closed form as a function of the Moebius parameters. Successively,
LMB2009 proposed a mixed parametrization, denoted by lambda, based on
marginal log-linear parameters such that graphical models for marginal
independence can be specified by setting to zero certain lambda terms. In this
framework, however, it is not possible to write the parameters of the
multinomial distribution as a function of lambda in closed form. In this paper, we
introduce a class of models for binary variables that we call the log-linear
Moebius models. A first feature of this class of models is that it includes, as a
special case, graphical models for marginal independence. The parameters of our
class of models, that we call gamma, are not a mixed parametrization and, in
fact, they are closely related to the Moebius parameters of DR2008 and allow us