  We propose sequential Monte Carlo (SMC) methods for sampling the posterior
distribution of state-space models under highly informative observation
regimes, a situation in which standard SMC methods can perform poorly. A
special case is simulating bridges between given initial and final values. The
basic idea is to introduce a schedule of intermediate weighting and resampling
times between observation times, which guide particles towards the final state.
This can always be done for continuous-time models, and may be done for
discrete-time models under sparse observation regimes; our main focus is on
continuous-time diffusion processes. The methods are broadly applicable in that
they support multivariate models with partial observation, do not require
simulation of the backward transition (which is often unavailable), and, where
possible, avoid pointwise evaluation of the forward transition. When simulating
bridges, the last cannot be avoided entirely without concessions, and we
suggest an epsilon-ball approach (reminiscent of Approximate Bayesian
Computation) as a workaround. Compared to the bootstrap particle filter, the
new methods deliver substantially reduced mean squared error in normalising
constant estimates, even after accounting for execution time. The methods are
demonstrated for state estimation with two toy examples, and for parameter
estimation (within a particle marginal Metropolis--Hastings sampler) with three
applied examples in econometrics, epidemiology and marine biogeochemistry.
