A Bayesian approach to covariance estimation and data fusion

In this paper, we address the fusion problem of two estimates, where the cross-correlation between the estimates is unknown. To solve the problem within the Bayesian framework, we assume that the covariance matrix has a prior distribution. We also assume that we know the covariance of each estimate, i.e., the diagonal block of the entire co-variance matrix (of the random vector consisting of the two estimates). We then derive the conditional distribution of the off-diagonal blocks, which is the cross-correlation of our interest. The conditional distribution happens to be the inverted matrix variate t-distribution. We can readily sample from this distribution and use a Monte Carlo method to compute the minimum mean square error estimate for the fusion problem. Simulations show that the proposed method works better than the popular covariance intersection method.