  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.
