Positive-definiteness in separable priors: effects on prior interpretability and inference

2026-05-21

AI Summary

This work investigates the unintended consequences of element-wise truncation in enforcing positive definiteness for symmetric positive definite matrix priors. While such truncation preserves positive definiteness, it systematically favors sparser structures under sparsity-inducing priors, thereby compromising prior interpretability, shrinkage properties, and posterior reliability. Through probabilistic truncation analysis, high-dimensional asymptotic theory, and sparse Bayesian inference, the study characterizes how truncation perturbs both dense and sparse priors and their marginal distributions. The key contribution lies in uncovering the structural bias induced by truncation in high dimensions and proposing a dimension-adaptive calibration strategy for prior parameters. This approach preserves positive definiteness while maintaining semantic consistency of the prior, substantially enhancing the stability and reliability of Bayesian modeling for sparse positive definite matrices.

Abstract

A popular class of priors for symmetric positive-definite matrices assumes independent entries and adds a truncation to ensure positive-definiteness. While conceptually simple and often computationally convenient, unless done carefully this truncation can have unintended effects. If the truncated prior or its margins are significantly different from their untruncated counterpart, then its interpretability may suffer, its shrinkage properties become harder to characterise, and posterior inference may be affected in unanticipated ways. We investigate the effect of the truncation both for dense and sparse matrices, and show how to set prior parameters such as the variance of off-diagonal entries such that said effect is mitigated as the matrix dimension grows. We pay particular attention to sparse inference where, unless prior parameters are set carefully, the truncated prior and hence its corresponding posterior assign systematically higher mass to sparser structures than the untruncated prior.

Problem

• positive-definiteness
• separable priors
• truncation effects
• prior interpretability
• sparse inference

Innovation

• positive-definiteness
• separable priors
• truncation effects
• sparse inference
• prior interpretability