Bayesian Markov-Switching Partial Reduced-Rank Regression

2025-12-19

AI Summary

Traditional reduced-rank regression assumes homogeneous, globally low-rank coefficient matrices and ignores potential group structures among response variables, thus failing to capture dynamic heterogeneity in temporal response relationships. To address this, we propose the Bayesian Markov-switching partial reduced-rank regression (BMSPRR) framework. BMSPRR adaptively partitions responses into two groups—“low-rank linear” and “Gaussian process nonlinear”—and jointly infers the time-varying group assignment, within-group ranks, model parameters, and regime-switching dynamics. We develop a partially collapsed Gibbs sampler for full Bayesian inference, circumventing dimension-changing sampling steps. Empirical evaluation on macroeconomic and commodity time-series data demonstrates that response grouping exhibits significant temporal variability and heterogeneous complexity across groups. BMSPRR achieves superior predictive accuracy and interpretability compared to static reduced-rank and fixed-group alternatives.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

Abstract

Reduced-Rank (RR) regression is a powerful dimensionality reduction technique but it overlooks any possible group configuration among the responses by assuming a low-rank structure on the entire coefficient matrix. Moreover, the temporal change of the relations between predictors and responses in time series induce a possibly time-varying grouping structure in the responses. To address these limitations, a Bayesian Markov-switching partial RR (MS-PRR) model is proposed, where the response vector is partitioned in two groups to reflect different complexity of the relationship. A extit{simple} group assumes a low-rank linear regression, while a extit{complex} group exploits nonparametric regression via a Gaussian Process. Differently from traditional approaches, group assignments and rank are treated as unknown parameters to be estimated. Then temporal persistence in the regression function is accounted for by a Markov-switching process that drives the changes in the grouping structure and model parameters over time. Full Bayesian inference is preformed via a partially collapsed Gibbs sampler, which allows uncertainty quantification without the need for trans-dimensional moves. Applications to two real-world macroeconomic and commodity data demonstrate the evidence of time-varying grouping and different degrees of complexity both across states and within each state.

Problem

• Addresses limitations of reduced-rank regression by modeling time-varying group structures in responses
• Proposes Bayesian Markov-switching model to partition responses into simple and complex groups
• Estimates unknown group assignments and ranks while accounting for temporal persistence via Markov-switching

Innovation

• Bayesian Markov-switching partial reduced-rank regression model
• Groups responses into simple low-rank and complex nonparametric regressions
• Estimates group assignments and rank via Gibbs sampler without trans-dimensional moves