We propose a methodology for Bayesian model determination in decomposable graphical gaussian models. To achieve this aim we consider a hyper inverse Wishart prior distribution on the concentration matrix for each given graph. To ensure compatibility across models, such prior distributions are obtained by marginalisation from the prior conditional on the complete graph. We explore alternative structures for the hyperparameters of the latter, and their consequences for the model. Model determination is carried out by implementing a reversible jump MCMC sampler. In particular, the dimension-changing move we propose involves adding or dropping an edge from the graph. We characterise the set of moves which preserve the decomposability of the graph, giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep. As state variable, we propose to use the incomplete variance-covariance matrix, containing only the elements for which the correspondi...

this paper we discuss both the theoretical design principles underlying the B-Course tool, and the pragmatic methods adopted in the implementation of the software

Given a set of possible models (e.g., Bayesian network structures) and a data sample, in the unsupervised model selection problem the task is to choose the most accurate model with respect to the domain joint probability distribution. In contrast to this, in supervised model selection it is a priori known that the chosen model will be used in the future for prediction tasks involving more \focused " predictive distributions. Although focused predictive distributions can be produced from the joint probability distribution by marginalization, in practice the best model in the unsupervised sense does not necessarily perform well in supervised domains. In particular, the standard marginal likelihood score is a criterion for the unsupervised task, and, although frequently used for supervised model selection also, does not perform well in such tasks. In this paper we study the performance of the marginal likelihood score empirically in supervised Bayesian network selection tasks by using a large number of publicly available classi cation data sets, and compare the results to those obtained by alternative model selection criteria, including empirical crossvalidation methods, an approximation of a supervised marginal likelihood measure, and a supervised version of Dawid's prequential (predictive sequential) principle. The results demonstrate that the marginal likelihood score does not perform well for supervised model selection, while the best results are obtained by using Dawid's prequential approach.

In this paper we study prequential model selection criteria in supervised learning domains. The main problem with this approach is the fact that the criterion is sensitive to the ordering the data is processed with. We discuss several approaches for addressing the ordering problem, and compare empirically their performance in real-world supervised model selection tasks. The empirical results demonstrate that with the prequential approach it is quite easy to find predictive models that are significantly more accurate classifiers than the models found by the standard unsupervised marginal likelihood criterion. The results also suggest that averaging over random orderings may be a more sensible strategy for solving the ordering problem than trying to find the ordering optimizing the prequential model selection criterion. 1

B-Course is a free web-based online data analysis tool, which allows the users to analyze their data for multivariate probabilistic dependencies. These dependencies are represented as Bayesian network models. In addition to this, B-Course also oers facilities for inferring certain type of causal dependencies from the data. The software uses a novel "tutorial style" userfriendly interface which intertwines the steps in the data analysis with support material that gives an informal introduction to the Bayesian approach adopted. Although the analysis methods, modeling assumptions and restrictions are totally transparent to the user, this transparency is not achieved at the expense of analysis power: with the restrictions stated in the support material, B-Course is a powerful analysis tool exploiting several theoretically elaborate results developed recently in the elds of Bayesian and causal modeling. B-Course can be used with most web-browsers (even Lynx), and the facilities include features such as automatic missing data handling and discretization, a exible graphical interface for probabilistic inference on the constructed Bayesian network models (for Java enabled browsers), automatic pretty-printed layout for the networks, exportation of the models, and analysis of the importance of the derived dependencies. In this paper we discuss both the theoretical design principles underlying the BCourse tool, and the pragmatic methods adopted in the implementation of the software.

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