Bayesian inference for Markov processes with diffusion and discrete components

Data arising in certain radio-tracking experiments consist of both a continuous spatial component and a discrete component related to behaviour. This leads naturally to stochastic models with a state space which is a product of continuous and discrete components. We consider a class of such models in continuous time, which can be thought of as diffusions in random environments. They are related to switching diffusion or hidden Markov models, but observations are made on both components at discrete time points, so that neither component is completely 'hidden'. We describe and illustrate an approach to fully Bayesian inference for these general models. The algorithm used is a hybrid Markov chain Monte Carlo method. The diffusion parameters, the environment parameters and the sample path of the environment process itself are updated separately, in sequence, and the individual steps are a mixture of Gibbs and random walk Metropolis-Hastings types. Some implementation and model checking issues are discussed, and an example using data arising from a radio-tracking experiment is described.