On Collapsed State-space Models and the Particle Marginal Metropolis-Hastings Sampler

Monte Carlo sampling of nonlinear state-space models is particularly difficult in circumstances where the transition density is not of closed form. This is common in the physical and biological sciences, where real phenomena demand modelling that accurately reproduces nonlinearity, chaos, mass conservation, multiple potentials and other behaviours that do not necessarily yield convenient analytical forms. In this work, we explore a new idea in the design of samplers for such models, collapsing out the state variables of a state-space model to leave only its noise terms. We then consider the design of proposal distributions over these noise terms, exploiting the independence and simple parametric forms prescribed to them in the prior structure of the model. The chosen vehicle for joint state and parameter estimation in these collapsed state-space models is the particle marginal Metropolis-Hastings (PMMH) sampler, from the family of particle Markov chain Monte Carlo (PMCMC). An auxiliary particle filter is used in the inner loop, with proposal distributions over noise terms tuned using the unscented Kalman filter. We conduct a thorough empirical investigation of the PMMH sampler in this context, including a look at the improvement obtained in acceptance rates, convergence rates and variability of the likelihood estimator. Case studies from the domain of marine biogeochemistry are used to demonstrate the ideas. We believe that these cases, exhibiting mass conservation and mild chaotic behaviour, are particularly challenging applications on which to exercise the PMCMC methodology.