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

TL;DR

This paper examines how truncation in priors for positive-definite matrices impacts interpretability and inference, offering guidelines to mitigate unintended effects especially in sparse matrix settings.

Contribution

It analyzes the effects of truncation on prior properties and provides methods to set parameters that reduce adverse impacts as matrix size increases.

Findings

Truncation can significantly alter prior interpretability and shrinkage.

Careful parameter setting mitigates effects of truncation in large matrices.

Truncated priors tend to favor sparser structures unless properly calibrated.

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.