  We present a general framework for Bayesian estimation of incompletely
observed multivariate diffusion processes. Observations are assumed to be
discrete in time, noisy and incomplete. We assume the drift and diffusion
coefficient depend on an unknown parameter. A data-augmentation algorithm for
drawing from the posterior distribution is presented which is based on
simulating diffusion bridges conditional on a noisy incomplete observation at
an intermediate time. The dynamics of such filtered bridges are derived and it
is shown how these can be simulated using a generalised version of the guided
proposals introduced in Schauer et al. (2016).
