Constrained Fiducial Inference for Gaussian Models

2026-02-11

AI Summary

This work addresses the challenge of parameter inference for non-i.i.d. Gaussian data, such as time series and spatial processes, by proposing a novel prior-free Markov chain Monte Carlo (MCMC) method. The approach uniquely integrates the Cayley transform with constrained generalized fiducial inference, leveraging a Cayley decomposition of the covariance matrix to construct a universal generative algorithm applicable to a broad class of parametric Gaussian models. Notably, the framework dispenses with both the i.i.d. assumption and Bayesian priors, enabling flexible and robust posterior-like inference. Simulation studies on MA(1) and Matérn covariance models demonstrate that the algorithm efficiently converges to the target confidence distribution, thereby validating its effectiveness and practical utility.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

We propose a new fiducial Markov Chain Monte Carlo (MCMC) method for fitting parametric Gaussian models. We utilize the Cayley transform to decompose the parametric covariance matrix, which in turn allows us to formulate a general data generating algorithm for Gaussian data. Leveraging constrained generalized fiducial inference, we are able to create the basis of an MCMC algorithm, which can be specified to parametric models with minimal effort. The appeal of this novel approach is the wide class of models which it permits, ease of implementation and the posterior-like fiducial distribution without the need for a prior. We provide background information for the derivation of the relevant fiducial quantities, and a proof that the proposed MCMC algorithm targets the correct fiducial distribution. We need not assume independence nor identical distribution of the data, which makes the method attractive for application to time series and spatial data. Well-performing simulation results of the MA(1) and Mat\'ern models are presented.

Problem

• Gaussian models
• fiducial inference
• covariance matrix
• time series
• spatial data

Innovation

• fiducial inference
• Cayley transform
• Gaussian models
• MCMC
• covariance decomposition