reversible jump on HMMs

Here is an email I received a few weeks ago about a paper written more than a decade ago in Glasgow with Tobias Rydén and Mike Titterington:

Sorry to bother you. I am a PhD student in economics. Recently, I am very interested in your paper “Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method”. I would like to use your method in estimating some regime-switching economic model. Unfortunately, I am not exactly understand your paper. Hence, I am writing to ask for your help. My questions are:

  1. A split or merge move is determined at the same time or sequentially? If the moves are determined at that same time, then accepting a split move implies that we can not accept a merge move any more in the same sweep. If the moves are determined sequentially, it means that we can accept a split move first, then accept a merge move in the same sweep. [Answer: First interpretation is correct. Except that the type of move is first selected at random, then only the corresponding move is generated and potentially accepted.]
  2.  In the paper, you discuss how to generate new transition probabilities in a split move in details. However, you did not discuss (probably, I am wrong) how to generate probabilities in each new state (series Zt in your paper).  Could you please tell me how to generate the series Zt? [Answer: check eqn (3).]
  3. My economic model is a multiple series (a vector hidden Markov model), will you refer me to some other papers for the vector model? [Answer: If the observed series is multidimensional, the extension is formally straightforward, if potentially prone to slow mixing and low acceptance rates. If the hidden Markov chain is multidimensional, I have not seen a version of reversible jump in this setting. Maybe an extension of the variational methods described in Ghahramani and Jordan would help.]

to which I replied that the questions showed a deep lack of understanding of what reversible jump is and that the PhD student should first check the literature, for instance the great intro paper by Charlie Geyer in Handbook of Markov chain Monte Carlo and then the original papers by Green (1995) and Richardson and Green (1997).

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