Practical Calculation of Non-Gaussian Multivariate Moments in Spatiotemporal Bayesian Maximum Entropy Analysis

During the past decade, the Bayesian maximum entropy (BME) approach has been used with considerable success in a variety of geostatistical applications, including the spatiotemporal analysis and estimation of multivariate distributions. In this work, we investigate methods for calculating the space/time moments of such distributions that occur in BME mapping applications, and we propose general expressions for non-Gaussian model densities based on Gaussian averages. Two explicit approximations for the covariance are derived, one based on leading-order perturbation analysis and the other on the diagrammatic method. The leading-order estimator is accurate only for weakly non-Gaussian densities. The diagrammatic estimator includes higher-order terms and is accurate for larger non-Gaussian deviations. We also formulate general expressions for Monte Carlo moment calculations including precision estimates. A numerical algorithm based on importance sampling is developed, which is computationally efficient for multivariate probability densities with a large number of points in space/time. We also investigate the BME moment problem, which consists in determining the general knowledge-based BME density from experimental measurements. In the case of multivariate densities, this problem requires solving a system of nonlinear integral equations. We refomulate the system of equations as an optimization problem, which we then solve numerically for a symmetric univariate pdf. Finally, we discuss theoretical and numerical issues related to multivariate BME solutions.