Summarizing Bayesian Nonparametric Mixture Posterior -- Sliced Optimal Transport Metrics for Gaussian Mixtures (2411.14674v5)

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

• The paper presents two novel metrics, Mix-SW and SMix-W, that improve efficiency in evaluating posterior mixing measures for BNP mixture models.
• It employs a decision-theoretic framework with sliced Wasserstein distances to minimize expected loss and optimize density estimation.
• Numerical experiments demonstrate that these methods outperform traditional techniques in balancing clustering performance and density accuracy.

An Examination of Bayesian Nonparametric Mixture Posterior Summarization Using Sliced Optimal Transport Metrics

This paper presents a novel methodology for summarizing posterior inference in Bayesian nonparametric (BNP) mixture models. It introduces a decision-theoretic approach focusing on density estimation via the mixing measure, distinguishing itself from typical methodologies which prioritize clustering through implied random partitions. A model-agnostic approach allows for dependencies within the generative model, while maintaining computational efficiency.

Major Contributions

The key contributions of this paper lie in the development of two novel implementations for Gaussian mixtures within the framework of sliced Wasserstein (SW) distances:

1. Mixed Sliced Wasserstein (Mix-SW): This metric applies generalized geodesic projections on the product of Euclidean space and the manifold of symmetric positive definite (SPD) matrices, leveraging the geometric properties of these spaces to retain useful information often discarded by simpler methods like vectorization.
2. Sliced Mixture Wasserstein (SMix-W): By leveraging linearity inherent in Gaussian mixtures, this distance provides an efficient projection methodology that balances computational complexity and geometric meaningfulness, outperforming traditional SW in certain scenarios.

Methodology

The authors employ a decision-theoretic framework to ascertain point estimates of posterior mixing measures by minimizing expected loss functions. By adopting optimal transport distances— notably SW due to its computational scalability—the method circumvents the common barriers related to ratio-based divergences like Kullback-Leibler divergence. Through Monte Carlo simulations, the expectations within the objective function are approximated, minimizing the computational overhead.

Theoretical and Practical Implications

The introduction of these new distance metrics, Mix-SW and SMix-W, expands the arsenal available for density estimation tasks in BNP models. Mix-SW particularly stands out by embedding Riemannian manifold properties into its projection methodology, showcasing potential applications in environments where geometric properties are pivotal. SMix-W capitalizes on Gaussian properties for more effective comparison of mixture densities, highlighting the continued evolution of transport-based distances in practical applications such as anomaly detection and data generation.

Numerical Validation

Through simulations, the authors validate the efficacy of the proposed distances for density and clustering tasks against conventional methods. Interestingly, Mix-SW and SMix-W report competitive clustering performance, effectively maintaining density estimations without materially compromising cluster recovery accuracy. The implicit suggestion here is a potential shift in priority towards more refined density estimations directly, with clustering addressed subsequently as an auxiliary output, rather than a primary target.

Future Directions

The potential of these methodologies indicates several avenues for future research. The paper notes the need for more sophisticated search algorithms to improve point estimates of mixing measures, as well as extensions beyond Gaussian mixtures to more comprehensive priors and richer mixture models. There's also the implicit potential for exploring other forms of geometric-aware projections that might contribute further improvements in density estimation across different model frameworks and applications.

In summary, by focusing on the mixing measure and employing novel transport distances, this paper paves the way for more efficient and effective density estimation in BNP mixture models, with promising implications for both theoretical understanding and practical application within AI methodologies.