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 corresponding element of the inverse is nonzero. This allows all computations to be performed locally, at the clique level, which is a clear advantage for the analysis of large and complex data-sets. Finally, the statistical and computational performance of the procedure is illustrated by means of both arti cial and real data-sets.
Summary. The major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of proposal. The first group of methods is based on an analysis of acceptance probabilities for jumps. Essentially, these methods involve a Taylor series expansion of the acceptance probability around certain canonical jumps and turn out to have close connections to Langevin algorithms. The second group of methods generalizes the reversible jump algorithm by using the so-called saturated space approach. These allow the chain to retain some degree of memory so that, when proposing to move from a smaller to a larger model, information is borrowed from the last time that the reverse move was performed. The main motivation for this paper is that, in complex problems, the probability that the Markov chain moves between such spaces may be prohibitively small, as the probability mass can be very thinly spread across the space. Therefore, finding reasonable jump proposals becomes extremely important. We illustrate the procedure by using several examples of reversible jump Markov chain Monte Carlo applications including the analysis of autoregressive time series, graphical Gaussian modelling and mixture modelling.
The late-2000s …nancial crisis has stressed the need of understanding the world …nancial system as a network of countries, where cross-border …nancial linkages play a fundamental role in the spread of systemic risks. Financial network models, that take into account the complex interrelationships between countries, seem to be an appropriate tool in this context. In this paper we propose to enrich the topological perspective of network models with a more structured statistical framework, that of graphical Gaussian models, which can be employed to accurately estimate the adjacency matrix, the main input for the estimation of the interconnections between di¤erent countries. We consider di¤erent types of graphical models: besides classical ones, we introduce Bayesian graphical models, that can take model uncertainty into account, and dynamic Bayesian graphical models, that provide a convenient framework to model temporal cross-border data, decomposing the model into autoregressive and contemporaneous networks. The paper shows how the application of the proposed models to the Bank of International Settlements locational banking statistics allows the identi…cation of four distinct groups of countries, that can be considered central in systemic risk contagion.
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