The Correct Birth/Death Jacobian for Mixture Models
Reversible jump Markov Chain Monte Carlo (RJMCMC)  is an extension of the Metropolis-Hastings algorithm that allows sampling from a distribution over models with potentially different numbers of parameters. In this post we are interested in determining the number of components to use when modeling data with a mixture model. The number of components corresponds to the dimension of the space we are walking through. The point of this post is to clear up a common error seen in the literature involving computing a Jacobian that arises in the algorithm.