This paper studies a problem of Bayesian parameter estimation for a sequence of scaled counting processes whose weak limit is a Brownian motion with an unknown drift. The main result of the paper is that the limit of the posterior distribution processes is, in general, not equal to the posterior distribution process of the mentioned Brownian motion with the unknown drift. Instead, it is equal to the posterior distribution process associated with a Brownian motion with the same unknown drift and a different standard deviation coefficient. The difference between the two standard deviation coefficients can be arbitrarily large. The characterization of the limit of the posterior distribution processes is then applied to a family of stopping time problems. We show that the proper way to find asymptotically optimal solutions to stopping time problems w.r.t.~the scaled counting processes is by looking at the limit of the posterior distribution processes rather than by the naive approach of looking at the limit of the scaled counting processes themselves. The difference between the performances can be arbitrarily large.