Completion problem with partial correlation vines

This paper extends the results in [D. Kurowicka, R.M. Cooke, A parametrization of positive definite matrices in terms of partial correlation vines, Linear Algebra Appl. 372 (2003) 225–251]. We show that a partial correlation vine represents a factorization of the determinant of the correlation matrix. We show that the graph of an incompletely specified correlation matrix is chordal if and only if it can be represented as an m-saturated incomplete vine, that is, an incomplete vine for which all edges corresponding to membership-descendents (m-descendents for short) of a specified edge are specified. This enables us to find the set of completions, and also the completion with maximal determinant for matrices corresponding to chordal graphs.