On the equivalence between standard and sequentially ordered hidden Markov models

TL;DR

This paper proves that sequentially ordered hidden Markov models are equivalent to standard HMMs in Bayesian terms, but not in Frequentist terms, highlighting differences in re-parametrisation concepts.

Contribution

It provides a formal proof of Bayesian equivalence between standard and sequentially ordered HMMs, clarifying their relationship.

Findings

Bayesian posterior distributions are equivalent for both models

Likelihood functions are incompatible between models in Frequentist perspective

Re-parametrisation concepts differ between Bayesian and Frequentist frameworks

Abstract

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.