# A Loss-Based Prior for Gaussian Graphical Models

## TL;DR

This paper introduces a new Bayesian prior for Gaussian graphical models that balances information loss and sparsity, improving graph selection in various applications.

## Contribution

It proposes a novel loss-based prior for the graph structure in Gaussian graphical models, incorporating information loss and sparsity considerations.

## Key findings

- The prior performs well on simulated data.
- It outperforms existing priors in real datasets.
- The method allows flexible calibration.

## Abstract

Gaussian graphical models play an important role in various areas such as genetics, finance, statistical physics and others. They are a powerful modelling tool which allows one to describe the relationships among the variables of interest. From the Bayesian perspective, there are two sources of randomness: one is related to the multivariate distribution and the quantities that may parametrise the model, the other has to do with the underlying graph, $G$, equivalent to describing the conditional independence structure of the model under consideration. In this paper, we propose a prior on G based on two loss components. One considers the loss in information one would incur in selecting the wrong graph, while the second penalises for large number of edges, favouring sparsity. We illustrate the prior on simulated data and on real datasets, and compare the results with other priors on $G$ used in the literature. Moreover, we present a default choice of the prior as well as discuss how it can be calibrated so as to reflect available prior information.