Inference and model choice for time-ordered hidden Markov models

We rewrite the system equation of a hidden Markov model so as to label the components by order of appearance, and make explicit the random behaviour of the number of components $m_t$. We argue that this reformulation is often the best way to achieve identifiability, as it facilitates the interpretation of the posterior density, and the estimation of the number of components that have appeared in a given sample. We develop a Sequential Monte Carlo algorithm for estimating the reformulated model, which relies on particle filtering and Gibbs sampling. Our algorithm has a computational cost similar to that of a MCMC sampler, and is much less likely to be affected by label switching, that is the possibility of getting trapped in a local mode of the posterior density. The extension to trans-dimensional priors is also considered. The approach is illustrated by real data examples, in particular the modelling of the instability of the relation between interest rates and the expected inflation rate in G7 countries.