Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals

Bayesian estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. Data-augmentation has been proposed for obtaining draws from the posterior distribution of the parameters. Within this approach, the discrete time observations are augmented with diffusion bridges connecting these observations. This poses two challenges: (i) efficiently generating diffusion bridges; (ii) if unknown parameters appear in the diffusion coefficient, then direct implementation of data-augmentation results in an induced Markov chain which is reducible. In this paper we show how both challenges can be addressed in continuous time (before discretisation) by using guided proposals. These are Markov processes with dynamics described by the stochastic differential equation of the diffusion process with an additional term added to the drift coefficient to guide the process to hit the right end point of the bridge. The form of these proposals naturally provides a mapping that decouples the dependence between the diffusion coefficient and diffusion bridge using the driving Brownian motion of the proposals. As the guiding term has a singularity at the right end point, care is needed when discretisation is applied for implementation purposes. We show that this problem can be dealt with by appropriately time changing and scaling of the guided proposal process. In two examples we illustrate the performance of the algorithms we propose. The second of these concerns a diffusion approximation of a chemical reaction network with a four-dimensional diffusion driven by an eight-dimensional Brownian motion.