  We consider an evolving system for which a sequence of observations is being
made, with each observation revealing additional information about current and
past states of the system. We suppose each observation is made without error,
but does not fully determine the state of the system at the time it is made.
  Our motivating example is drawn from invasive species biology, where it is
common to know the precise location of invasive organisms that have been
detected by a surveillance program, but at any time during the program there
are invaders that have not been detected.
  We propose a sequential importance sampling strategy to infer the state of
the invasion under a Bayesian model of such a system. The strategy involves
simulating multiple alternative states consistent with current knowledge of the
system, as revealed by the observations. However, a difficult problem that
arises is that observations made at a later time are invariably incompatible
with previously simulated states. To solve this problem, we propose a two-step
iterative process in which states of the system are alternately simulated in
accordance with past observations, then corrected in light of new observations.
We identify criteria under which such corrections can be made while maintaining
appropriate importance weights.
