Graphical posterior predictive classification: Bayesian model averaging with particle Gibbs

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 J. Olsson, T. Pavlenko, and F. L. Rios [Electron. J. Statist. 13 (2019), no. 2, 2865–2897] for posterior sampling from decomposable graphical models which utilizes the so-called Christmas tree algorithm of J. Olsson, T. Pavlenko, and F. L. Rios [Stat. Comput. 32 (2022), no. 5, Paper No. 80, 18] as proposal kernel. We also derive a strong hyper Markov law which we call the hyper normal Wishart law that allows 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.