Probability Based Independence Sampler for Bayesian Quantitative Learning in Graphical Log-Linear Marginal Models (original) (raw)
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2005
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Bayesian estimation of Gaussian graphical models has proven to be challenging because the conjugate prior distribution on the Gaussian precision matrix, the G-Wishart distribution, has a doubly intractable partition function. Recent developments provide a direct way to sample from the G-Wishart distribution, which allows for more efficient algorithms for model selection than previously possible. Still, estimating Gaussian graphical models with more than a handful of variables remains a nearly infeasible task. Here, we propose two novel algorithms that use the direct sampler to more efficiently approximate the posterior distribution of the Gaussian graphical model. The first algorithm uses conditional Bayes factors to compare models in a Metropolis-Hastings framework. The second algorithm is based on a continuous time Markov process. We show that both algorithms are substantially faster than state-of-theart alternatives. Finally, we show how the algorithms may be used to simultaneously estimate both structural and functional connectivity between subcortical brain regions using resting-state fMRI. * mhinne@cs.ru.nl
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Conditional independence models are deflned by a set of conditional in- dependence restrictions and play an important role in many statistical applications, especially, but not only, graphical modeling. In this paper we identify a subclass of these models which are hierarchical marginal log-linear, as deflned by Bergsma and Rudas (2002a). Such models are smooth, which implies the applicability of standard asymptotic theory and simplifles interpretation. Furthermore, we give a marginal log- linear parameterization and a minimal speciflcation of the models in the subclass, which implies the applicability of standard methods to compute maximum likelihood estimates and simplifles the calculation of the degrees of freedom for the model. We illustrate the utility of our results by applying them to certain block recursive Markov models associated with chain graphs.