Model-free Bayesian Clustering and Multimodal Posterior Summarisation

Qing Li; Sara Wade

stat.MLOct 9, 2026

Model-free Bayesian Clustering and Multimodal Posterior Summarisation

Authors: Qing Li, Sara Wade

Organizations: School of Mathematics University of Edinburgh

Abstract

We develop a model-free, noise-aware Bayesian clustering framework that constructs a martingale posterior over clusterings from a predictive Bayesian perspective. This avoids explicit likelihood-prior model specification, expensive MCMC sampling, and mode-seeking variational approximations, while accommodating irregular cluster shapes and enabling parallel generation of posterior draws. To interpret the posterior over high-dimensional, noise-aware clusterings, we introduce a Wasserstein-type summarization method. Using a novel noise-aware loss, this approach compresses the large collection of posterior draws into a small set of weighted representatives, capturing distinct cluster structures and noise assignments. Further, we marginalise the uncertainty in the density level, avoiding reliance on a specific choice. We establish theoretical properties of the proposed loss and guarantees for the level-marginal posterior approximation. Results demonstrate the framework's effectiveness in capturing clustering and noise uncertainty.

Figures & tables

Figure 1: A model-free noise-aware clustering framework with posterior summarisation.

Figure 2: Point-estimate clusterings for the ring (top) and four-rectangle (bottom) examples. Colours indicate active clusters and grey crosses are inactive observations; dashed black boundaries show the true cluster regions. $\hat{K}$ is the estimated number of clusters. BNPmix ( Corradin et al., 2021 ) shows the fitted Gaussian components. We present the Bayes estimate of the proposed MDBC under both the VI IA and BI IA losses in ( 3 ).

Figure 3: Illustration of predictive-resampling with density draws (blue lines) mapped to clustering draws by level-set rule.

	Ring $(n=250,\ K=1,\ \mathrm{Noise}=90)$					Four rectangles $(n=480,\ K=4,\ \mathrm{Noise}=86)$
Metric	$k$ -means	DBSCAN	BNPmix	MDBC ${}_{\text{V}}$	MDBC ${}_{\text{B}}$	$k$ -means	DBSCAN	BNPmix	MDBC ${}_{\text{V}}$	MDBC ${}_{\text{B}}$
$\hat{K}$	1	3	7	1	3	3	7	8	4	4
$\hat{\text{N}}\text{oise}$	0	44	0	73	88	0	79	0	68	68
$\widetilde{\mathrm{VI}}_{\mathrm{IA}}\downarrow$	0.239	0.225	0.428	0.082	0.178	0.174	0.050	0.158	0.024	0.024
$\widetilde{\mathrm{BI}}_{\mathrm{IA}}\downarrow$	0.360	0.396	0.684	0.124	0.292	0.273	0.062	0.236	0.037	0.037
$\mathrm{ARI}\uparrow$	0.000	0.183	0.019	0.558	0.324	0.483	0.859	0.604	0.917	0.917

Table 1: Point-estimate clustering accuracy.

\hat{K}

and

\hat{\text{N}}\text{oise}

are the estimated number of clusters and inactive data points, respectively. Lower is better for the normalized

\widetilde{\mathrm{VI}}_{\mathrm{IA}}

and

\widetilde{\mathrm{BI}}_{\mathrm{IA}}

; higher is better for

\mathrm{ARI}

. Point estimates of MDBC use the same posterior draws and differ in the loss ( V for VI IA and B for BI IA ).

Figure 4: (a) VI IA compression loss for $L=1$ and increasing separation $d$ . (b) Elbow plot for choosing $L$ ; the dashed line marks the selected $L=2$ . (c)–(d) The two particles at $d=2.5$ . Grey crosses are inactive observations.

Figure 5: Robustness to increasing clutter. (a) One-particle $\mathrm{VI}_{\mathrm{IA}}$ compression loss and its decomposition into active–inactive and within-active clustering components. (b) Normalised $\widetilde{\mathrm{VI}}_{\mathrm{IA}}$ against the ground-truth clustering (lower is better). (c) Estimated inactive fraction, with the dotted line indicating the true clutter fraction. In (b)–(c), the black line denotes the $L=1$ point estimate of the proposed method, while the shaded fan summarises the distribution across posterior sub-partition draws, with shading indicating central posterior coverage.

Figure 6: Harbour data. (a) Satellite map of the site. (b) VI IA point estimate; colours denote cluster membership and grey crosses are clutter. (c) Expected VI IA contribution around point estimate; marker shape indicates active/inactive status and darker colours indicate greater uncertainty.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Illustration of chaining in the $\delta$ -graph and its suppression by reciprocal- $k$ NN connectivity. Two dense clusters are separated by a single bridge point (green star) lying above the density threshold $\lambda$ ; grey points lie below the threshold and are inactive. (a) In the $\delta$ -graph $G_{\delta,\lambda}(f)$ , the bridge is connected to active observations in both clusters, merging them into a single connected component. (b) In the reciprocal- $k$ NN graph $G^{\mathrm{rkNN}}_{k,\lambda}(f)$ with $k=5$ , the bridge selects nearby observations, but they do not select it in return. The bridge therefore remains isolated and no longer connects the two dense components.

Figure 8: Hasse diagram of the sub-partition lattice. Moving upward corresponds either to merging two active blocks or to activating one previously inactive observation as a singleton active block. The bottom element is the empty sub-partition $\varnothing$ .

Panel of Figure 9 — Figure 9: Data-driven calibration of the density level. Top: ring example. Bottom: four-rectangle example. In each case, the left panel shows the sorted $3$ -nearest-neighbour distances, with the orange markers denoting the detected knee $\varepsilon$ , defined as the point of maximum perpendicular distance from the chord (grey lines). The middle panel transfers the same empirical rank $\hat{\nu}$ to the posterior median MPD density, giving $\lambda=q_{\hat{\nu}}\{\hat{f}(x_{i})\}$ . The right panel shows the resulting reference active and inactive sets under $\lambda$ . Green dotted lines indicate the true component boundaries for visual reference and are not used in the calibration.

Figure 10: Density estimates for the ring (top) and four-rectangle (bottom) examples. Columns show the MPD posterior mean, the BNPmix posterior mean, Gaussian DPMM posterior means under concentration priors $\alpha\sim\mathrm{Ga}(2,4)$ and $\alpha\sim\mathrm{Ga}(2,1)$ , and the mclust plug-in density estimate fitted by EM with BIC-based model selection. Dashed curves indicate the underlying high-density structures. Panels within each row share a common colour scale.

Figure 11: Dataset and ground truth clustering in the separation experiment. Red and blue points denote the two Gaussian components and grey crosses denote background clutter.

Figure 12: Posterior structural behaviour across additional Monte Carlo realisations. Posterior probabilities of a merged configuration ( $K=1$ ), a genuine two-cluster split ( $K=2$ , $n_{\min}\geq 20$ ), and a small-fragment configuration ( $K=2$ , $n_{\min}<20$ ) as the separation $d$ increases. Solid curves show the median across realisations, shaded regions show the full range, and circles show individual realisations.

Figure 13: VI IA uncertainty across additional Monte Carlo realisations. Total $L=1$ VI IA compression loss (left), its active/inactive component (centre), and the restricted-VI structural component on the common active set (right), as a function of separation $d$ . Boxes show the interquartile range, whiskers the full range, and circles the individual Monte Carlo realisations.

Figure 14: Clustering results under increasing clutter contamination, $\epsilon\in\{0.1,0.2,0.3,0.4\}$ . Columns correspond to increasing clutter levels, while rows show the ground truth, proposed method, DBSCAN, gap-selected $k$ -means, and BNPmix, respectively. The component observations are held fixed while clutter points are added sequentially across columns. Grey crosses denote inactive or unassigned observations.

Figure 15: Selection of the number of WASABI-SuP particles for the harbour data. Total approximation loss against the number of particles $L$ for BI IA (left) and VI IA (right). The dotted vertical lines indicate the values selected by the elbow criterion, $L=2$ for BI IA and $L=3$ for VI IA . Absolute loss magnitudes are not comparable across the two loss functions.

Figure 16: Three-particle VI IA WASABI-SuP summary for the harbour data.

Figure 17: Two-particle BI IA WASABI-SuP summary for the harbour data. Crosses denote inactive observations.

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Scalable Posterior Uncertainty for Flexible Density-Based Clustering

Nicola Bariletto, Stephen G. Walker

Department of Statistics and Data Sciences, UT Austin

Density-Based Clustering Clustering Uncertainty Quantification Bayesian Inference

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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.

Abstract

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