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

2024-11-22

AI Summary

This paper addresses the longstanding limitation in posterior summarization for nonparametric Bayesian mixture models—where inference has predominantly focused on random partition point estimates, neglecting direct inference on the mixing measure itself. We propose a decision-theoretic framework that prioritizes the mixing measure as the primary inferential target. Methodologically, we introduce, for the first time, a model-agnostic variant of the sliced Wasserstein distance, integrated with generalized geodesic projection and optimization on the symmetric positive-definite matrix manifold; leveraging the linear structure of Gaussian mixing measures, our approach delivers coherent point estimates of the mixing measure, density function, and random partition simultaneously. Compared to conventional paradigms, our method preserves statistical validity under complex dependency structures, substantially improves geometric coherence and computational efficiency in posterior summarization, and unifies support for both density estimation and clustering inference.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Relational Probabilistic ModelsSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalizationSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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

Problem

• Summarize posterior inference in nonparametric Bayesian mixtures
• Estimate mixing measure using sliced Wasserstein distance
• Develop model-agnostic approach for Gaussian mixtures

Innovation

• Uses sliced Wasserstein distance for mixing measures
• Applies geodesic projections on Euclidean space
• Leverages linearity of Gaussian mixture measures