# A permutation-based Bayesian approach for inverse covariance estimation

## TL;DR

This paper introduces a permutation-based Bayesian method for estimating inverse covariance matrices in high-dimensional Gaussian graphical models, addressing the challenge of unknown variable ordering and demonstrating improved accuracy and stability.

## Contribution

It proposes a novel permutation-based Bayesian approach that is order-invariant and outperforms existing methods in inverse covariance estimation.

## Key findings

- Method achieves smaller variability in estimates.
- Establishes posterior convergence rates.
- Outperforms existing approaches in simulations.

## Abstract

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.