\name{I.g}
\alias{I.g}

\title{ Normalizing constant of G-Wishart distribution }

\description{
Calculates the normalizing constant of G-Wishart distribution based on Monte Carlo method, developed by Atay-Kayis and Massam (2005).
}

\usage{
I.g( G, b = 3, D = diag( ncol(G) ), mc = 100 )
}

\arguments{
  \item{G}{
Adjacency matrix corresponding to the graph structure. It is an upper triangular matrix in which 
\eqn{g_{ij}=1}{gij = 1} if there is a link between notes \eqn{i}{i} and \eqn{j}{j}, 
otherwise \eqn{g_{ij}=0}{gij = 0}.
}
  \item{b}{ The degree of freedom for G-Wishart distribution, \eqn{W_G(b, D)}. The default value is 3.}
  
  \item{D}{
The positive definite \eqn{(p \times p)} "scale" matrix for G-Wishart distribution, 
\eqn{W_G(b,D)}. The default is identity matrix.
}
  \item{mc}{The number of iteration for the Monte Carlo approximation. The default value is 100.}
}

\details{
Normalizing constant approximation using Monte Carlo method for a G-Wishart distribution, \eqn{K \sim W_G(b, D)}, with density:

  \deqn{Pr(K) = \frac{1}{I(b, D)} |K| ^ {(b - 2) / 2} \exp \left\{- \frac{1}{2} \mbox{trace}(K \times D)\right\}.}
}

\value{ The normalizing constant of G-Wishart distribution. }

\references{
Atay-Kayis, A. and H. Massam (2005). A monte carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models, \emph{Biometrika}, 92(2):317-335

Mohammadi, A. and E. Wit (2015). Bayesian Structure Learning in Sparse Gaussian Graphical Models, \emph{Bayesian Analysis}, 10(1):109-138
}

\author{ Abdolreza Mohammadi and Ernst Wit }

\examples{
G <- matrix( c(0,0,1,
               0,0,1,
		               0,0,0), 3, 3, byrow = TRUE )
		                
# matrix G is an adjacency matrix of a graph with 3 nodes and 2 links
I.g( G, b = 3, D = diag(3) )
}