Bayesian Multiple Multivariate Density-Density Regression

2026-01-06

AI Summary

This work proposes the first distribution-to-distribution regression framework to address the challenge of modeling multivariate density-valued responses against multivariate density-valued predictors. The method constructs fitted distributions via sliced Wasserstein barycenters (SWB) and formulates inference as an inverse SWB problem within a generalized Bayesian paradigm, where discrepancies are measured by the sliced Wasserstein distance. By combining a differentiable SWB approximation with a smooth reparameterization respecting simplex constraints, the approach enables efficient gradient-based MCMC sampling and likelihood-free uncertainty quantification. Theoretical analysis establishes perturbation stability of SWB, sample complexity bounds, and posterior consistency. Applied to single-cell population data, the method not only accurately predicts distributional responses but also yields an interpretable, sparse intercellular communication network through posterior barycentric weights.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We propose the first approach for multiple multivariate density-density regression (MDDR), making it possible to consider the regression of a multivariate density-valued response on multiple multivariate density-valued predictors. The core idea is to define a fitted distribution using a sliced Wasserstein barycenter (SWB) of push-forwards of the predictors and to quantify deviations from the observed response using the sliced Wasserstein (SW) distance. Regression functions, which map predictors'supports to the response support, and barycenter weights are inferred within a generalized Bayes framework, enabling principled uncertainty quantification without requiring a fully specified likelihood. The inference process can be seen as an instance of an inverse SWB problem. We establish theoretical guarantees, including the stability of the SWB under perturbations of marginals and barycenter weights, sample complexity of the generalized likelihood, and posterior consistency. For practical inference, we introduce a differentiable approximation of the SWB and a smooth reparameterization to handle the simplex constraint on barycenter weights, allowing efficient gradient-based MCMC sampling. We demonstrate MDDR in an application to inference for population-scale single-cell data. Posterior analysis under the MDDR model in this example includes inference on communication between multiple source/sender cell types and a target/receiver cell type. The proposed approach provides accurate fits, reliable predictions, and interpretable posterior estimates of barycenter weights, which can be used to construct sparse cell-cell communication networks.

Problem

• density-density regression
• multivariate density
• multiple predictors
• Wasserstein barycenter
• Bayesian inference

Innovation

• Sliced Wasserstein Barycenter
• Density-Density Regression
• Generalized Bayes
• Inverse Barycenter Problem
• Differentiable Approximation