Model-free Bayesian Clustering and Multimodal Posterior Summarisation

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.

From neuroscience and genomics to systems biology and ecology, researchers rely on clustering similarity data to uncover modular structure. Most methods work in two steps: first reduce noise, then cluster. The first step runs without regard to the second: it forces arbitrary choices, discards useful information, and can impose structure where none exists. Here we bring the two steps together. Using established Bayesian community detection methods, we let one minimum description length criterion choose both how much to sparsify the data and how to cluster it. This one-step approach keeps all the samples for inference, removes the arbitrary threshold choice, and returns a single cluster when there is no real structure. On synthetic data, our approach recovers clusters under more noise and with fewer samples than standard methods. On gene co-expression data from the human brain, it finds more coherent and more stable clusters than WGCNA, the standard tool in the field. Uniting sparsification and clustering in a single model-selection step provides a principled, noise-resistant way to uncover modular structure in similarity data across fields.

The empirical success of diffusion models in generative modelling has motivated theoretical work, including quantitative error bounds and qualitative analyses that characterise the different phases of denoising. We bring these two areas together by studying the adaptivity of diffusion models to the structured geometry of multimodal high-dimensional data that consists of multiple clusters in $\mathbb{R}^D$, each with its own low-dimensional structure, and inter-cluster separation depending on $D$. We employ $K$-mixture Gaussian distributions as a canonical framework to capture this geometry and establish two theoretical results. First, we interpret denoising as a dynamical Bayesian classifier: the mixture score is a posterior-weighted average of cluster-wise scores, and we show that, with high probability, the posterior class probabilities concentrate on a single cluster once the signal-to-noise ratio reaches the scale $Θ(\log (KD)/D)$. Second, by separately analysing the denoising process in its mixing and cluster-commitment phases, we prove that the KL error bound depends linearly on the maximum intrinsic dimension of a cluster, up to a logarithmic factor, even when $K$ grows polynomially with $D$. This improves on ambient-dimensional bounds and extends existing low-dimensional adaptivity analyses to multimodal distributions with heterogeneous, approximately low-rank covariances.