Reversible jump and the label switching problem in hidden Markov models

• Published in: Journal of statistical planning and inference Vol. 139; no. 7; pp. 2305 - 2315
• Main Author: Spezia, Luigi
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.07.2009; Elsevier
• Subjects: Classification; Clustering; Distribution theory; Exact sciences and technology; General topics; Identifiability; Markov mixture models; Markov processes; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Trans-dimensional MCMC; Identifiability; Markov mixture models; Trans-dimensional MCMC; Clustering; Classification; Invariant; Rejection sampling; Statistical distribution; Iteration; Stochastic method; Markov chain; Reversible jump; Approximation theory; Bayes estimation; Mixed distribution; Monte Carlo method; Fitting; Prior distribution; Permutation; Statistical estimation; Statistical decision; Markov model; Gibbs sampling; Algorithm; Statistical method; Numerical analysis; Hidden Markov models; Hidden Markov model
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2008.10.016

Reversible jump Markov chain Monte Carlo (RJMCMC) algorithms can be efficiently applied in Bayesian inference for hidden Markov models (HMMs), when the number of latent regimes is unknown. As for finite mixture models, when priors are invariant to the relabelling of the regimes, HMMs are unidentifiable in data fitting, because multiple ways to label the regimes can alternate during the MCMC iterations; this is the so-called label switching problem. HMMs with an unknown number of regimes are considered here and the goal of this paper is the comparison, both applied and theoretical, of five methods used for tackling label switching within a RJMCMC algorithm; they are: post-processing, partial reordering, permutation sampling, sampling from a Markov prior and rejection sampling. The five strategies we compare have been proposed mostly in the literature of finite mixture models and only two of them, i.e. rejection sampling and partial reordering, have been presented in RJMCMC algorithms for HMMs. We consider RJMCMC algorithms in which the parameters are updated by Gibbs sampling and the dimension of the model changes in split-and-merge and birth-and-death moves. Finally, an example illustrates and compares the five different methodologies.