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Abstract:
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Continuous-time diffusions (stochastic differential equations) are popular models in fields like finance, biology and chemistry. These serve as priors over hidden trajectories, and given noisy observations, one is interested in the posterior distribution over paths and process parameters. Inference is complicated by the intractable transition probabilities, making MCMC 'doubly-intractable'. Often one uses approximate MCMC schemes that discretize time; Beskos and Roberts 2005, developed an exact algorithm with no such error. Based on an accept-reject scheme, they evaluate Brownian proposals using an auxiliary Poisson process. While exact, large observation intervals and informative measurements can lead to high rejection rates and slow mixing. We develop a Gibbs sampler that ameliorates some of these issues. Treating the auxiliary Poisson process as a random discretization of time, we alternately sample a new discretization given the diffusion path, and a new path given the discretization. We show the first step is easy using Poisson thinning, and the second involves standard discrete-time techniques. We compare our ideas with existing algorithms on applications in finance.
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