Summarizing Nonparametric Bayesian Mixture Posteriors – Sliced Optimal Transport Metrics for Gaussian Mixtures

Existing methods to summarize posterior inference for mixture models focus on identifying a point estimate of the implied random partition for clustering and validating it using clustering-based loss functions (Wade and Ghahramani, 2018; Dahl et al., 2022). We propose a novel approach for summarizing posterior inference in nonparametric Bayesian mixture models, prioritizing estimation of the mixing measure as an inference target. Moreover, we propose to validate the estimation of the partition via a new perspective which is based on the implied density estimate. One of the key features is the model-agnostic nature of the approach, which remains valid under arbitrarily complex dependence structures in the underlying sampling model. Using a decision-theoretic framework, the proposed methods identify a point estimate using loss functions defined as discrepancies between mixing measures. Estimating the mixing measure then implies inference on the mixture density and the random partition. To define a discrepancy between mixing measures we exploit the discrete nature of the mixing measure and use a version of sliced Wasserstein distance. We introduce two variants for Gaussian mixtures. The first, mixed sliced Wasserstein, applies generalized geodesic projections on a product of Euclidean space and the manifold of symmetric positive definite matrices. The second, sliced mixture Wasserstein, leverages the linearity of Gaussian mixture measures to define a projection.