We introduce a novel framework for uncertainty quantification in clustering that combines martingale posterior distributions with density-based clustering. Unlike classical model-based approaches, which define clusters at the latent level of a mixture model, we treat clusters as explicit functionals of the data-generating density, without assuming any specific parametric form. To characterize density uncertainty, we obtain martingale posterior samples via a predictive resampling scheme driven by model score evaluations. This allows us to leverage state-of-the-art differentiable density estimators, such as normalizing flows, making density resampling efficient in large-scale settings and fully parallelizable on modern GPU hardware. Martingale posterior samples of the clustering structure are then obtained by applying density-based clustering to the density draws, enabling principled inference on any clustering-related quantity. Casting the inference target as a density functional further enables a rigorous theoretical analysis of the procedure's convergence properties. We apply our methodology to image and single-cell RNA sequencing data, demonstrating the computational efficiency afforded by its GPU compatibility as well as its ability to recover meaningful clustering structures, with associated uncertainty, across diverse domains.

翻译：我们提出了一种新颖的聚类不确定性量化框架，该框架将鞅后验分布与基于密度的聚类相结合。与经典模型方法（即在混合模型的潜变量层面定义簇）不同，我们将簇视为数据生成密度的显式泛函，且不假设任何特定的参数形式。为了表征密度不确定性，我们通过基于模型分数评估的预测重采样方案获取鞅后验样本。这使我们能够利用最先进的可微密度估计器（如归一化流），从而在大规模场景下实现高效的密度重采样，并在现代GPU硬件上完全并行化。随后，通过将基于密度的聚类应用于密度采样结果，可获得聚类结构的鞅后验样本，从而实现对任意聚类相关量的原理性推断。将推断目标视为密度泛函，还可对方法的收敛性质进行严格的理论分析。我们将该方法应用于图像和单细胞RNA测序数据，展示了其GPU兼容性带来的计算效率，以及在不同领域中恢复具有不确定性的有意义聚类结构的能力。