Bayesian inference for spatio-temporal hidden Markov models using the exchange algorithm

Spatio-temporal hidden Markov models are extremely difficult to estimate because their latent joint distributions are available only in trivial cases. In the estimation phase, these latent distributions are usually substituted with pseudo-distributions, which could affect the estimation results, in particular in the presence of strong dependence between the latent variables. In this work, we propose a spatio-temporal hidden Markov model where the latent process is an extension of the autologistic model. We show how inference can be carried out in a Bayesian framework using an approximate exchange algorithm, which circumvents the impractical calculations of the normalizing constants that arise in the model. Our proposed method leads to a Markov chain Monte Carlo sampler that targets the correct posterior distribution of the model and not a pseudo-posterior. In addition, we develop a new initialization approach for the approximate exchange method, reducing the computational time of the algorithm. An extensive simulation study shows that the approximate exchange algorithm generally outperforms the pseudo-distribution approach, yielding more accurate parameter estimates. Finally, the proposed methodology is applied to a real-world case study analyzing rainfall levels across Italian regions over time.