  We study a classical Bayesian statistics problem of sequentially testing the
sign of the drift of an arithmetic Brownian motion with the $0$-$1$ loss
function and a constant cost of observation per unit of time for general prior
distributions. The statistical problem is reformulated as an optimal stopping
problem with the current conditional probability that the drift is non-negative
as the underlying process. The volatility of this conditional probability
process is shown to be non-increasing in time, which enables us to prove
monotonicity and continuity of the optimal stopping boundaries as well as to
characterize them completely in the finite-horizon case as the unique
continuous solution to a pair of integral equations. In the infinite-horizon
case, the boundaries are shown to solve another pair of integral equations and
a convergent approximation scheme for the boundaries is provided. Also, we
describe the dependence between the prior distribution and the long-term
asymptotic behaviour of the boundaries.
