On Bayesian Graphical Model Determination

Akademisk avhandling
som för avläggande av filosofie doktorsexamen
vid Stockholms universitet
offentligen försvaras i
De Geersalen, Geovetenskapens hus, Frescati
måndagen den 31 januari 2000 kl 10.00


Jukka Corander
fil lic

Statistiska institutionen
Stockholms universitet

A graphical model specifies a graph representation of the independence structure of a multivariate distribution, where nodes represent variables and edges association between variables.
This thesis introduces methodology for determination of graphical models for multivariate distributions within the exponential family. Model determination is understood in the present context as quantification of the uncertainty about the association structure, given empirical observations. Only models with symmetric associations between variables are considered. The distributions investigated are multinomial, multinormal and conditional Gaussian (CG) distributions. Local graphical models which generalize the graphical loglinear models for multinomial distributions are introduced. These models allow conditional associations to be absent locally, in parts of the sample space.
A unifying theme is that the models are represented in terms of affine restrictions to the parameters of a regular exponential model. All introduced methods are applicable to the complete class of graphical models, consisting of both decomposable and non-decomposable models. Various real data sets investigated earlier in the graphical modeling literature are used to illustrate the methods.
Two different measures of model uncertainty are considered: the posterior probability and the relative expected utility of a model. Posterior probabilities are estimated by a Markov chain Monte Carlo sampling method. The other measure of model uncertainty is derived in a decision theoretic framework under reference priors for the model parameters.
The expected logarithmic utility of a model is decomposed into predictive performance and relative cost. The predictive performance is measured by posterior expectation of the negative entropy of the distribution induced by a graphical model. This expectation has an analytic expression for decomposable models, while a simulation consistent estimate can be obtained for non-decomposable models. The expected logarithmic utility is asymptotically equivalent to the Schwarz criterion under a certain cost function.

Key words: Bayesian inference, Entropy, Exponential models, Graphical models, Logarithmic utility.

ISBN 91-7265-036-2