  We study finite horizon optimal switching problems for hidden Markov chain
models under partially observable Poisson processes. The controller possesses a
finite range of strategies and attempts to track the state of the unobserved
state variable using Bayesian updates over the discrete observations. Such a
model has applications in economic policy making, staffing under variable
demand levels and generalized Poisson disorder problems. We show regularity of
the value function and explicitly characterize an optimal strategy. We also
provide an efficient numerical scheme and illustrate our results with several
computational examples.
