# Bayesian Analysis of Nonparanormal Graphical Models Using Rank-Likelihood

## TL;DR

This paper introduces a Bayesian method for nonparanormal graphical models that uses a rank-likelihood to avoid modeling transformation functions, employing a horseshoe prior and Gibbs sampling for efficient inference.

## Contribution

It proposes a novel Bayesian approach utilizing rank-likelihood for nonparanormal models, with theoretical consistency and practical efficiency demonstrated.

## Key findings

- The method achieves accurate graph structure recovery in simulations.
- Posterior consistency is established for the precision matrix.
- The approach performs well on real data applications.

## Abstract

Gaussian graphical models, where it is assumed that the variables of interest jointly follow a multivariate normal distribution with a sparse precision matrix, have been used to study intrinsic dependence among variables, but the normality assumption may be restrictive in many settings. A nonparanormal graphical model is a semiparametric generalization of a Gaussian graphical model for continuous variables where it is assumed that the variables follow a Gaussian graphical model only after some unknown smooth monotone transformation. We consider a Bayesian approach for the nonparanormal graphical model using a rank-likelihood which remains invariant under monotone transformations, thereby avoiding the need to put a prior on the transformation functions. On the underlying precision matrix of the transformed variables, we consider a horseshoe prior on its Cholesky decomposition and use an efficient posterior Gibbs sampling scheme. We present a posterior consistency result for the precision matrix based on the rank-based likelihood. We study the numerical performance of the proposed method through a simulation study and apply it on a real dataset.