This paper examines the ordering of measurement updates for a general Bayesian inference problem and its impact on the estimation of the posterior distribution. The approach used compares the expected improvement to the posterior from various types of potential measurements, taking into account the current estimated prior but not the actual measurements, to determine the optimal measurement to perform and/or incorporate. The expected improvement is quantified using both an entropy and a covariance-based measure, each of which is further approximated for computational expedience. Compared to a random ordering of measurements, the posterior is observed to converge more quickly, resulting in a significant improvement in performance.