Summarizing Bayesian Nonparametric Mixture Posterior -- Sliced Optimal Transport Metrics for Gaussian Mixtures

TL;DR

This paper introduces a model-agnostic, decision-theoretic method for summarizing Bayesian nonparametric mixture posteriors by estimating the mixing measure using sliced Wasserstein distances, applicable to complex dependence structures.

Contribution

It proposes a novel, model-agnostic approach for summarizing posterior distributions in nonparametric Bayesian mixture models, focusing on estimating the mixing measure via sliced Wasserstein metrics.

Findings

The method effectively estimates the mixing measure in Gaussian mixtures.

It remains valid under complex dependence structures.

Two variants of the approach improve computational efficiency.

Abstract

Existing methods to summarize posterior inference for mixture models focus on identifying a point estimate of the implied random partition for clustering, with density estimation as a secondary goal (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 (or mixture) as an inference target. 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, our method identifies a point estimate by minimizing posterior expected loss. A loss function is defined as a discrepancy between mixing measures. Estimating the mixing measure implies inference on the mixture density and the random partition. Exploiting the discrete nature of the mixing measure, we use a version of sliced Wasserstein distance. We introduce two specific variants for Gaussian mixtures. The first, mixed sliced Wasserstein, applies generalized geodesic projections on the product of the Euclidean space and the manifold of symmetric positive definite matrices. The second, sliced mixture Wasserstein, leverages the linearity of Gaussian mixture measures for efficient projection