Consistency of Graphical Model-based Clustering: Robust Clustering using Bayesian Spanning Forest
Yu Zheng, Leo L. Duan, Arkaprava Roy
TL;DR
This work analyzes a graphical model-based clustering framework built on Bayesian spanning forest (BSF), showing that, under mild conditions, the integrated posterior on node partitions concentrates on the true clustering even when the data-generating process differs from the specified graph-based model. The authors prove clustering consistency for both fixed and diverging numbers of clusters, derive a misclassification-rate bound when the true number of clusters is known, and introduce a refinement technique to bound posterior ratios. They specialize the results to Gaussian-BSF and extend to general object-valued distributions on metric spaces, with explicit conditions on separation and tail behavior. The findings support the robustness of BSF as an alternative to mixture models, and the developed determinant-bounding and refinement methods provide tools potentially useful beyond this specific setting.
Abstract
Mixture model-based framework is very popular for statistical inference on clustering. 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 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. Our result holds for both cases when the number of clusters is fixed or diverging as the sample size increases, and further provides a statistical upper bound of the misclassification rate. These theoretical results are encouraging developments for the model-based clustering literature, demonstrating the use of graphical models as a robust alternative to mixture models.