The optimal detection procedure for detecting changes in independent and identically distributed sequences (i.i.d.) in a Bayesian setting was derived by Shiryaev in the nineteen sixties. However, the analysis of the performance of this procedure in terms of the average detection delay and false alarm probability has been an open problem. In this paper, we investigate the performance of Shiryaev's procedure in an asymptotic setting where the false alarm probability goes to zero. The asymptotic study is performed not only in. the i.d.d. case where the Shiryaev's procedure is optimal but also in a general, non-i.i.d. case. In the latter case, we show that Shiryaev's procedure is asymptotically optimum under mild conditions. We also show that the two popular non-Bayesian detection procedures, namely the Page and Shiryaev-Roberts-Pollak procedures, are not optimal (even asymptotically) under the Bayesian criterion. The results of this study are shown to be especially important in studying the asymptotics of decentralized quickest change detection procedures.