A note on Metropolis-Hasting for sampling across mixed spaces
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We are concerned with the Metropolis-Hastings algorithm for sampling across so-called mixed spaces. The most important example of a mixed space is a space that consists of real vectors of different lengths. Sampling within this space is called trans dimensional sampling and enjoys an enormous popularity across many statistical communities. However, the ubiquitous literature is completely divided over the abilities of the Metropolis-Hastings algorithm in these scenarios. This lead to the success of the reversible jump algorithm as the quasi gold standard. Unfortunately, the actual contribution of reversible jump is consistently misunderstood. Thus, this paper gives a dense overview of the theory that builds the mathematical foundations of sampling within mixed spaces, discusses a changepoint example and finally clears up any misunderstandings.
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