Informative Distance-Based Priors for Correlation Matrices Centred on a Target Reference

2026-07-21

AI Summary

This work addresses the challenge of specifying priors for Bayesian modeling of high-dimensional correlation matrices, whose parameter space forms a curved manifold with dimensionality growing quadratically in the number of variables. The authors propose an exponential decay prior based on the Fisher arc-length distance that shrinks toward a user-specified target correlation matrix and incorporates conditional independence constraints induced by a graph structure via the Cholesky decomposition of the inverse correlation matrix. Unlike conventional approaches limited to shrinkage toward the identity matrix, this method accommodates arbitrary target structures and naturally interprets complexity penalization as a geometric penalty for deviation from the target. It supports correlations of any sign and integrates seamlessly with graphical models. The resulting prior is well-defined for any positive rate parameter, reduces to an unstructured prior under a complete graph, and admits a direct sampling algorithm for prior predictive checks and sensitivity analysis, implemented in the graphpcor package.

Abstract

Specifying a prior over the space of correlation matrices is a persistent challenge in Bayesian analysis. The space is a curved manifold whose dimension grows quadratically with the number of variables, making substantive prior beliefs difficult to encode.\\ We propose a distance-based prior that assigns mass decaying exponentially in the Fisher arc-length distance from a user-specified reference correlation matrix, enabling shrinkage toward any target correlation structure rather than being confined to the identity matrix. Formally, this is constructed as a Penalised Complexity prior, but its interpretation shifts accordingly: unless the chosen target represents a structurally simpler state, the shrinkage penalises deviation rather than complexity in the usual sense. To accommodate conditional independence constraints, we introduce a parameterisation that constructs the correlation matrix via the Cholesky factor of the inverse correlation matrix with respect to a user-supplied graph, thereby reducing the number of free parameters from one per variable pair to one per graph edge. The prior is proper for every positive value of its rate parameter, accommodates correlations of either sign under any graph structure, and reduces to a fully unstructured prior when the graph is complete. A direct sampling algorithm is provided, enabling prior predictive checks and sensitivity analysis, implemented within the \texttt{graphpcor} package.

Problem

• correlation matrices
• Bayesian prior
• distance-based prior
• curved manifold
• conditional independence

Innovation

• distance-based prior
• correlation matrix
• Penalised Complexity prior
• Cholesky parameterisation
• graphical models