  We consider the problem of inference for nonlinear, multivariate diffusion
processes, satisfying It\^o stochastic differential equations (SDEs), using
data at discrete times that may be incomplete and subject to measurement error.
Our starting point is a state-of-the-art correlated pseudo-marginal
Metropolis-Hastings algorithm, that uses correlated particle filters to induce
strong and positive correlation between successive likelihood estimates.
However, unless the measurement error or the dimension of the SDE is small,
correlation can be eroded by the resampling steps in the particle filter. We
therefore propose a novel augmentation scheme, that allows for conditioning on
values of the latent process at the observation times, completely avoiding the
need for resampling steps. We integrate over the uncertainty at the observation
times with an additional Gibbs step. Connections between the resulting
pseudo-marginal scheme and existing inference schemes for diffusion processes
are made, giving a unified inference framework that encompasses Gibbs sampling
and pseudo marginal schemes. The methodology is applied in three examples of
increasing complexity. We find that our approach offers substantial increases
in overall efficiency, compared to competing methods.
