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

av

**Jukka Corander**

fil lic

Statistiska institutionen

Stockholms universitet

**ABSTRACT**

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