  In this study, we present a multi-class graphical Bayesian predictive
classifier that incorporates the uncertainty in the model selection into the
standard Bayesian formalism. For each class, the dependence structure
underlying the observed features is represented by a set of decomposable
Gaussian graphical models. Emphasis is then placed on the Bayesian model
averaging which takes full account of the class-specific model uncertainty by
averaging over the posterior graph model probabilities. An explicit evaluation
of the model probabilities is well known to be infeasible. To address this
issue, we consider the particle Gibbs strategy of Olsson et al. (2018b) for
posterior sampling from decomposable graphical models which utilizes the
Christmas tree algorithm of Olsson et al. (2018a) as proposal kernel. We also
derive a strong hyper Markov law which we call the hyper normal Wishart law
that allow to perform the resultant Bayesian calculations locally. The proposed
predictive graphical classifier reveals superior performance compared to the
ordinary Bayesian predictive rule that does not account for the model
uncertainty, as well as to a number of out-of-the-box classifiers.
