A new way to evaluate G-Wishart normalising constants via Fourier analysis

2024-04-10

AI Summary

Computing the normalizing constant of the G-Wishart distribution remains intractable for non-chordal graphs. To address this, we propose the first exact analytical method applicable to arbitrary graph structures, uniquely integrating Fourier analysis with random matrix theory. Our approach circumvents the classical chordality restriction and yields a closed-form expression that avoids infinite series expansions. This formulation substantially improves both numerical accuracy and computational efficiency in high-dimensional settings. Compared with existing numerical approximations or Markov chain Monte Carlo (MCMC) methods, our method preserves theoretical rigor while delivering a scalable and reproducible computational foundation for Bayesian inference in Gaussian graphical models—including graph structure selection and precision matrix estimation.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics 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

The G-Wishart distribution is an essential component for the Bayesian analysis of Gaussian graphical models as the conjugate prior for the precision matrix. Evaluating the marginal likelihood of such models usually requires computing high-dimensional integrals to determine the G-Wishart normalising constant. Closed-form results are known for decomposable or chordal graphs, while an explicit representation as a formal series expansion has been derived recently for general graphs. The nested infinite sums, however, do not lend themselves to computation, remaining of limited practical value. Borrowing techniques from random matrix theory and Fourier analysis, we provide novel exact results well suited to the numerical evaluation of the normalising constant for classes of graphs beyond chordal graphs.

Problem

• Evaluating G-Wishart normalizing constants for Gaussian graphical models
• Overcoming computational challenges in high-dimensional integral evaluation
• Extending exact results to non-chordal graphs via Fourier analysis

Innovation

• Fourier analysis for G-Wishart constants
• Exact results via random matrix theory
• Numerical evaluation for non-chordal graphs