Scalable Posterior Uncertainty for Flexible Density-Based Clustering

Density-based clustering is a well-established framework that defines clusters through topological or geometric features of the data-generating density, yielding flexible procedures that have proven effective on large and complex datasets. We address the problem of obtaining scalable Bayesian-style uncertainty quantification by combining density-based clustering with martingale posterior distributions. Density uncertainty is first quantified through a fully parallelized predictive resampling scheme driven by model score functions, naturally accommodating modern differentiable density estimators such as normalizing flows. Posterior samples of the clustering are then obtained by applying density-based clustering to each random density sampled from the martingale posterior, enabling inference on any clustering-related quantity. Casting the estimand as a density functional also enables a rigorous frequentist convergence analysis. We illustrate the method through applications to image and single-cell RNA sequencing data, where it recovers meaningful clustering structures, including the assignment of previously unseen data and biologically interpretable cell grouping, while running in minutes on a single GPU.