Marginal reversible jump Markov chain Monte Carlo with application to motor unit number estimation

Motor unit number estimation (MUNE) is a method which aims to provide a quantitative indicator of progression of diseases that lead to a loss of motor units, such as motor neurone disease. However the development of a reliable, repeatable and fast real-time MUNE method has proved elusive hitherto. Previously, a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm has been implemented to produce a posterior distribution for the number of motor units using a Bayesian hierarchical model that takes into account biological information about motor unit activation. However this approach can be unreliable for some datasets since it can suffer from poor cross-dimensional mixing. The focus is on improved inference by marginalising over latent variables to create the likelihood. More specifically, the emphasis is on how this marginalisation can improve the RJMCMC mixing and that alternative approaches that utilise the likelihood (e.g. DIC) can be investigated. For this model the marginalisation is over latent variables which, for a larger number of motor units, is an intractable summation over all combinations of a set of latent binary variables whose joint sample space increases exponentially with the number of motor units. A tractable and accurate approximation for this quantity is provided and also other approximations based on Monte Carlo estimates that can be incorporated into RJMCMC are investigated.