## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 31, Number 3 (2017), 569-617.

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

Kushal Kr. Dey and Sourabh Bhattacharya

#### 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.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 31, Number 3 (2017), 569-617.

**Dates**

Received: March 2015

Accepted: June 2016

First available in Project Euclid: 22 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1503388830

**Digital Object Identifier**

doi:10.1214/16-BJPS325

**Mathematical Reviews number (MathSciNet)**

MR3693982

**Zentralblatt MATH identifier**

1378.60100

**Keywords**

Additive transformation diffusion limit high dimension optimal scaling random walk transformation-based Markov Chain Monte Carlo

#### Citation

Dey, Kushal Kr.; Bhattacharya, Sourabh. A brief tutorial on transformation based Markov Chain Monte Carlo and optimal scaling of the additive transformation. Braz. J. Probab. Stat. 31 (2017), no. 3, 569--617. doi:10.1214/16-BJPS325. https://projecteuclid.org/euclid.bjps/1503388830

#### Supplemental materials

- Supplement to “A brief tutorial on transformation based Markov Chain Monte Carlo and optimal scaling of the additive transformation”. Additional details are provided in this supplementary material, whose sections and figures have the prefix “S-” when referred to in this article. Briefly, in Section S-1, we provide details on computational efficiency of TMCMC. Specifically, we demonstrate with an experiment the superior computational speed of additive TMCMC in comparison with RWM, particularly in high dimensions. In Section S-2 we discuss, with appropriate experiments, the necessity of optimal scaling in additive TMCMC, while in Sections S-3 and S-4 we delve into the robustness issues associated with the scale choices of additive TMCMC and RWM. In Section S-5, we include brief discussions of adaptive versions of RWM and TMCMC. Moreover, the proofs of all our technical results are provided in Sections S-6 and S-7 of the supplement.Digital Object Identifier: doi:10.1214/16-BJPS325SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.