Consistency of Graphical Model-based Clustering: Robust Clustering using Bayesian Spanning Forest

For statistical inference on clustering, the mixture model-based framework is very popular. On the one hand, the model-based framework is convenient for producing probabilistic estimates of cluster assignments and uncertainty. On the other hand, the specification of a mixture model is fraught with the danger of misspecification that could lead to inconsistent clustering estimates. Graphical model-based clustering takes a different model specification strategy, in which the likelihood treats the data as arising dependently from a disjoint union of component graphs. To counter the large uncertainty of the graph, recent work on Bayesian spanning forest proposes using the integrated posterior of the node partition (marginalized over the latent edge distribution) to produce probabilistic estimates for clustering. Despite the strong empirical performance, it is not yet known whether the clustering estimator is consistent, especially when the data-generating mechanism is different from the specified graphical model. This article gives a positive answer in the asymptotic regime: when the data arise from an unknown mixture distribution, under mild conditions, the posterior concentrates on the ground-truth partition, producing correct clustering estimates including the number of clusters. This theoretical result is an encouraging development for the robust clustering literature, demonstrating the use of graphical models as a robust alternative to mixture models in model-based clustering.