Loglinear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical loglinear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in contingency tables. The approach that is taken is graphical and draws on analogies with multivariate Gaussian models for marginal independence. For the graphical model representation we use bidirected graphs, which are in the tradition of path diagrams. We show how the models can be parameterized in a simple fashion, and how maximum likelihood estimation can be performed by using a version of the iterated conditional fitting algorithm. Finally we consider combining these models with symmetry restrictions.
Graphical Gaussian process models for highly multivariate spatial data
Summary For multivariate spatial Gaussian process models, customary specifications of crosscovariance functions do not exploit relational intervariable graphs to ensure processlevel conditional independence between the variables. This is undesirable, especially in highly multivariate settings, where popular crosscovariance functions, such as multivariate Matérn functions, suffer from a curse of dimensionality as the numbers of parameters and floatingpoint operations scale up in quadratic and cubic order, respectively, with the number of variables. We propose a class of multivariate graphical Gaussian processes using a general construction called stitching that crafts crosscovariance functions from graphs and ensures processlevel conditional independence between variables. For the Matérn family of functions, stitching yields a multivariate Gaussian process whose univariate components are Matérn Gaussian processes, and which conforms to processlevel conditional independence as specified by the graphical model. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate the utility of the graphical Matérn Gaussian process to jointly model highly multivariate spatial data using simulation examples and an application to airpollution modelling.
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 Award ID(s):
 1915803
 NSFPAR ID:
 10334291
 Date Published:
 Journal Name:
 Biometrika
 ISSN:
 00063444
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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