Iterative Conditional Fitting for Estimation of a Covariance Matrix with Zeros

We consider estimation of the covariance matrix of a random vector under the constraint that certain elements in the covariance matrix are zero. Assuming that the random vector follows a multivariate normal distribution, in which case the model is also known as covariance graph model, we present a new algorithm for maximum likelihood estimation of the covariance matrix with zero pattern. We give our new algorithm the name Iterative Conditional Fitting since in each step of the procedure, a conditional distribution is estimated, subject to constraints, while a marginal distribution is held fixed. This approach is in duality to the well-known iterative proportional fitting algorithm, in which marginal distributions are fitted while conditional distributions are held fixed. We show that Iterative Conditional Fitting can be implemented using least squares computations and we establish the convergence properties of the algorithm.