A brief tutorial on transformation based Markov Chain Monte Carlo and optimal scaling of the additive transformation

Abstract

We consider the recently introduced Transformation-based Markov Chain Monte Carlo (TMCMC) (Stat. Methodol. 16 (2014) 100–116), a methodology that is designed to update all the parameters simultaneously using some simple deterministic transformation of a one-dimensional 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 one-dimensional 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.