  We show that the optimal decision policy for several types of Bayesian
sequential detection problems has a threshold switching curve structure on the
space of posterior distributions. This is established by using lattice
programming and stochastic orders in a partially observed Markov decision
process (POMDP) framework. A stochastic gradient algorithm is presented to
estimate the optimal linear approximation to this threshold curve. We
illustrate these results by first considering quickest time detection with
phase-type distributed change time and a variance stopping penalty. Then it is
proved that the threshold switching curve also arises in several other Bayesian
decision problems such as quickest transient detection, exponential delay
(risk-sensitive) penalties, stopping time problems in social learning, and
multi-agent scheduling in a changing world. Using Blackwell dominance, it is
shown that for dynamic decision making problems, the optimal decision policy is
lower bounded by a myopic policy. Finally, it is shown how the achievable cost
of the optimal decision policy varies with change time distribution by imposing
a partial order on transition matrices.
