Use of auxiliary variables for generating proposal variables within a Metropolis-Hastings setting has been suggested in many different settings. This has in particular been of interest for simulation from complex distributions such as multimodal distributions or in transdimensional approaches. For many of these approaches, the acceptance probabilities that are used turn up somewhat magic and different proofs for their validity have been given in each case.
In this paper I will present a general framework for construction of acceptance probabilities in auxiliary variable proposal generation. In addition to demonstrate the similarities between many of the proposed algorithms in the literature, the framework also demonstrate that there is a great flexibility in how to construct such acceptance probabilities, in addition to the flexibility in how to construct the proposals. With this flexibility, alternative acceptance probabilities are suggested. Some numerical experiments are also reported.