Moment Matching Based Conjugacy Approximation for Bayesian Ranking and Selection

We study the conjugacy approximation method in the context of Bayesian ranking and selection with unknown correlations. Under the assumption of normal-inverse-Wishart prior distribution, the posterior distribution remains a normal-inverse-Wishart distribution thanks to the conjugacy property when all alternatives are sampled at each step. However, this conjugacy property no longer holds if only one alternative is sampled at a time, an appropriate setting when there is a limited budget on the number of samples. We propose two new conjugacy approximation methods based on the idea of moment matching. Both of them yield closed-form Bayesian prior updating formulas. This updating formula can then be combined with the knowledge gradient algorithm under the "value of information" framework. We conduct computational experiments to show the superiority of the proposed conjugacy approximation methods, including applications in wind farm placement and computer model calibration.