On the equivalence between standard and sequentially ordered hidden Markov models

Chopin (2007) introduced a sequentially ordered hidden Markov model, for which states are ordered according to their order of appearance, and claimed that such a model is a re-parametrisation of a standard Markov model. This note gives a formal proof that this equivalence holds in Bayesian terms, as both formulations generate equivalent posterior distributions, but does not hold in Frequentist terms, as both formulations generate incompatible likelihood functions. Perhaps surprisingly, this shows that Bayesian re-parametrisation and Frequentist re-parametrisation are not identical concepts.