Transdimensional transformation based Markov chain Monte Carlo

In this article, we propose a novel and general dimension-hopping MCMC methodology that can update all the parameters as well as the number of parameters simultaneously using simple deterministic transformations of some low-dimensional (often one-dimensional) random variable. This methodology, which has been inspired by the recent Transformation based MCMC (TMCMC) for updating all the parameters simultaneously in general fixed-dimensional set-ups using low-dimensional random variables, facilitates great speed in terms of computation time and provides high acceptance rates, thanks to the low-dimensional random variables which effectively reduce the dimension dramatically. Quite importantly, our transformation based approach provides a natural way to automate the move-types in the variable dimensional problems. We refer to this methodology as Transdimensional Transformation based Markov Chain Monte Carlo (TTMCMC). We develop the theory of TTMCMC, illustrating it with gamma and normal mixtures with unknown number of components, for both simulated and real data sets. Comparisons with RJMCMC demonstrates far superior performance of TTMCMC in terms of mixing, acceptance rate, computational speed and automation. Furthermore, we demonstrate good performance of TTMCMC in multivariate normal mixtures, even for dimension as large as 20. To our knowledge, there exists no application of RJMCMC for such high-dimensional mixtures. Further, we propose a novel methodology to summarize the posterior, providing a way to obtain the mode of the posterior distribution of the densities and the associated highest posterior density credible regions. Based on our method we also propose a criterion to assess convergence of variable-dimensional algorithms. These methods of summarization and convergence assessment are applicable to general problems, not just to mixtures.