A Brief Tutorial on Transformation Based Markov Chain Monte Carlo and Optimal Scaling of the Additive Transformation

TL;DR

This paper provides a tutorial on Transformation-based Markov Chain Monte Carlo (TMCMC), analyzes its diffusion limits, and demonstrates its advantages over traditional methods like random walk Metropolis through theoretical results and empirical studies.

Contribution

It introduces the additive TMCMC method, explores its theoretical properties including optimal scaling and acceptance rates, and compares its performance to existing MCMC algorithms.

Findings

Optimal acceptance rate for additive TMCMC is 0.439.

Additive TMCMC has advantages over random walk Metropolis.

Theoretical analysis of diffusion limits and scaling for additive TMCMC.

Abstract

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.