Joint density of correlations in the correlation matrix with chordal sparsity patterns

We extend the methodology of generating the random correlation matrix of Joe (2006) and Lewandowski et al. (2009) by introducing a partial correlation expansion of the determinant of a correlation matrix which is more general than the partial correlations on a regular vine used in Lewandowski et al. (2009). This generalization allows us to formulate the partial correlation expansion of determinant for a correlation matrix with a chordal sparsity pattern. For such a partially specified correlation matrix we find a uniform density of unspecified correlations. This leads to a closed form formula for the volume of the space of correlation matrices with specified correlations corresponding to a chordal graph. We present an algorithm to generate uniformly a random correlation matrix with a chordal sparsity pattern.