**by Jukka Corander**

**Research Report
1999:5**

Department of Statistics, Stockholm University, S-106 91 Stockholm, Sweden

**Abstract**

Bayesian model determination for multinomial and multinormal data in the complete class of graphical models is considered using a decision theoretic framework. The complete class contains both decomposable and non-decomposable graphical models. A utility measure based on a logarithmic score function is introduced under reference priors for the model parameters. The expected logarithmic utility of a model is decomposed into its predictive performance and relative complexity. The predictive performance is measured by the expected posterior entropy, which is shown to have an analytic expression for decomposable models when certain reference priors are used. For non-decomposable models, a simulation consistent estimate of the expected entropy can be obtained. Asymptotic equivalence between the expected logarithmic utility and the logarithm of the marginal likelihood of a model is briefly discussed. Both real and simulated data sets are used to illustrate the introduced methodology.

**Key words**: Bayesian
model determination, Entropy, Graphical models, Laplace approximation,
Multinomial distribution, Multinormal distribution, Reference analysis,
Schwarz criterion, Utility.

*Last update: 2000-02-15/CE*