  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.
