Bayesian Inference in Nonparanormal Graphical Models

TL;DR

This paper introduces a Bayesian method for nonparanormal graphical models, allowing for flexible dependence modeling among variables after unknown smooth transformations, with efficient inference and consistency guarantees.

Contribution

It develops a Bayesian approach using B-spline priors for transformations and spike-and-slab priors for the precision matrix, enabling flexible, consistent inference in nonparanormal models.

Findings

Method performs well in simulations.

Achieves posterior consistency.

Successfully applied to real data.

Abstract

Gaussian graphical models have been used to study intrinsic dependence among several variables, but the Gaussianity assumption may be restrictive in many applications. A nonparanormal graphical model is a semiparametric generalization for continuous variables where it is assumed that the variables follow a Gaussian graphical model only after some unknown smooth monotone transformations on each of them. We consider a Bayesian approach in the nonparanormal graphical model by putting priors on the unknown transformations through a random series based on B-splines where the coefficients are ordered to induce monotonicity. A truncated normal prior leads to partial conjugacy in the model and is useful for posterior simulation using Gibbs sampling. On the underlying precision matrix of the transformed variables, we consider a spike-and-slab prior and use an efficient posterior Gibbs sampling scheme. We use the Bayesian Information Criterion to choose the hyperparameters for the spike-and-slab prior. We present a posterior consistency result on the underlying transformation and the precision matrix. We study the numerical performance of the proposed method through an extensive simulation study and finally apply the proposed method on a real data set.