A Point Mass Proposal Method for Bayesian State-Space Model Fitting

State-space models (SSMs) are often used to model time series data where the observations depend on an unobserved latent process. However, inference on the process parameters of an SSM can be challenging, especially when the likelihood of the data given the parameters is not available in closed-form. We focus on the problem of model fitting within a Bayesian framework, for which existing approaches include Markov chain Monte Carlo (MCMC) using Bayesian data augmentation, sequential Monte Carlo approximation, and particle MCMC algorithms, which combine sequential Monte Carlo approximations and MCMC steps. However, these different methods can be inefficient when sample impoverishment occurs during the sequential Monte Carlo approximation and/or when the MCMC algorithm mixes poorly. We propose the use of deterministic hidden Markov models (HMMs) to provide an efficient MCMC with data augmentation approach, imputing the latent states within the algorithm. Our approach deterministically approximates the SSM by a discrete HMM, which is subsequently used as an MCMC proposal distribution for the latent states in Metropolis-within-Gibbs steps. We demonstrate that the algorithm provides an efficient alternative method for state-space models with near-chaotic behaviour.