  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.
