  We consider the recently introduced Transformation-based Markov Chain Monte
Carlo (TMCMC) (Dutta and Bhattacharya (2014)), a methodology that is designed
to update all the parameters simultaneously using some simple deterministic
transformation of a onedimensional random variable drawn from some arbitrary
distribution on a relevant support. The additive transformation based TMCMC is
similar in spirit to random walk Metropolis, except the fact that unlike the
latter, additive TMCMC uses a single draw from a onedimensional proposal
distribution to update the high-dimensional parameter. In this paper, we first
provide a brief tutorial on TMCMC, exploring its connections and contrasts with
various available MCMC methods. Then we study the diffusion limits of additive
TMCMC under various set-ups ranging from the product structure of the target
density to the case where the target is absolutely continuous with respect to a
Gaussian measure; we also consider the additive TMCMC within Gibbs approach for
all the above set-ups. These investigations lead to appropriate scaling of the
one-dimensional proposal density. We also show that the optimal acceptance rate
of additive TMCMC is 0.439 under all the aforementioned set-ups, in contrast
with the well-established 0.234 acceptance rate associated with optimal random
walk Metropolis algorithms under the same set-ups. We also elucidate the
ramifications of our results and clear advantages of additive TMCMC over random
walk Metropolis with ample simulation studies and Bayesian analysis of a real,
spatial dataset with which 160 unknowns are associated.
