Application of Bayesian item selection criteria in computerized adaptive testing might result in improvement of bias and MSE of the ability estimates. The question remains how to apply Bayesian item selection criteria in the context of constrained adaptive testing, where large numbers of specifications have to be taken into account in the item selection process. The Shadow Test Approach is a general purpose algorithm for administering constrained CAT. In this paper it is shown how the approach can be slightly modified to handle Bayesian item selection criteria. No differences in performance were found between the shadow test approach and the modified approach. In a simulation study of the LSAT, the effects of Bayesian item selection criteria are illustrated. The results are compared to item selection based on Fisher Information. General recommendations about the use of Bayesian item selection criteria are provided. (Contains 4 figures.)