Optimal covariance selection for estimation using graphical models

We consider a problem encountered when trying to estimate a Gaussian random field using a distributed estimation approach based on Gaussian graphical models. Because of constraints imposed by estimation tools used in Gaussian graphical models, the a priori covariance of the random field is constrained to embed conditional independence constraints among a significant number of variables. The problem is, then: given the (unconstrained) a priori covariance of the random field, and the conditional independence constraints, how should one select the constrained covariance, optimally representing the (given) a priori covariance, but also satisfying the constraints? In 1972, Dempster provided a solution, optimal in the maximum likelihood sense, to the above problem. Since then, many works have used Dempster´s optimal covariance, but none has addressed the issue of suitability of this covariance for Bayesian estimation problems. We prove that Dempster´s covariance is not optimal in most minimum mean squared error (MMSE) estimation problems. We also propose a method for finding the MMSE optimal covariance, and study its properties. We then illustrate the analytical results via a numerical example, that demonstrates the estimation performance advantage gained by using the optimal covariance vs Dempster´s covariance. The numerical example also shows that, for the particular estimation scenario examined, Dempster´s covariance violates the necessary conditions for optimality.