A note on the Metropolis-Hastings acceptance probabilities for mixture spaces

This work is driven by the ubiquitous dissent over the abilities and contributions of the Metropolis-Hastings and reversible jump algorithm within the context of trans dimensional sampling. We demystify this topic by taking a deeper look into the implementation of Metropolis-Hastings acceptance probabilities with regard to general mixture spaces. Whilst unspectacular from a theoretical point of view, mixture spaces gave rise to challenging demands concerning their effective exploration. An often applied but not extensively studied tool for transitioning between distinct spaces are so-called translation functions. We give an enlightening treatment of this topic that yields a generalization of the reversible jump algorithm and unveils another promising translation technique. Furthermore, by reconsidering the well-known Metropolis within Gibbs paradigm, we come across a dual strategy to develop Metropolis-Hastings samplers. We underpin our findings and compare the performance of our approaches by means of a change point example. Thereafter, in a more theoretical context, we revitalize the somewhat forgotten concept of maximal acceptance probabilities. This allows for an interesting classification of Metropolis-Hastings algorithms and gives further advice on their usage. A review of some errors in reasoning that have led to the aforementioned dissent concludes this paper.