In 1960s Shiryaev developed Bayesian theory of change detection in independent and identically distributed (i.i.d.) sequences. In Shiryaev's classical setting the goal is to minimize an average detection delay under the constraint imposed on the average probability of false alarm. Recently, Tartakovsky and Veeravalli (2005) developed a general Bayesian asymptotic change-point detection theory (in the classical setting) that is not limited to a restrictive i.i.d. assumption. It was proved that Shiryaev's detection procedure is asymptotically optimal under traditional average false alarm probability constraint, assuming that this probability is small. In the present paper, we consider a less conventional approach where the constraint is imposed on the global, supremum false alarm probability. An asymptotically optimal Bayesian change detection procedure is proposed and thoroughly evaluated for both i.i.d. and non-i.i.d. models when the global false alarm probability approaches zero.