## Brazilian Journal of Probability and Statistics

### Transdimensional transformation based Markov chain Monte Carlo

#### Abstract

Variable dimensional problems, where not only the parameters, but also the number of parameters are random variables, pose serious challenge to Bayesians. Although in principle the Reversible Jump Markov Chain Monte Carlo (RJMCMC) methodology is a response to such challenges, the dimension-hopping strategies need not be always convenient for practical implementation, particularly because efficient “move-types” having reasonable acceptance rates are often difficult to devise.

In this article, we propose and develop 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 Transformation based MCMC (TMCMC) of (Stat. Mehodol. (2014) 16 100–116), facilitates great speed in terms of computation time and provides reasonable acceptance rates and mixing properties. Quite importantly, our approach provides a natural way to automate the move-types in variable dimensional problems. We refer to this methodology as Transdimensional Transformation based Markov Chain Monte Carlo (TTMCMC). Comparisons with RJMCMC in gamma and normal mixture examples demonstrate 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.

As by-products of our effort on the development of TTMCMC, we propose a novel methodology to summarize the posterior distributions of the mixture densities, 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.

#### Article information

Source
Braz. J. Probab. Stat., Volume 33, Number 1 (2019), 87-138.

Dates
Accepted: October 2017
First available in Project Euclid: 14 January 2019

https://projecteuclid.org/euclid.bjps/1547456488

Digital Object Identifier
doi:10.1214/17-BJPS380

Mathematical Reviews number (MathSciNet)
MR3898723

Zentralblatt MATH identifier
07031065

#### Citation

Das, Moumita; Bhattacharya, Sourabh. Transdimensional transformation based Markov chain Monte Carlo. Braz. J. Probab. Stat. 33 (2019), no. 1, 87--138. doi:10.1214/17-BJPS380. https://projecteuclid.org/euclid.bjps/1547456488

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#### Supplemental materials

• Supplement to “Transdimensional transformation based Markov chain Monte Carlo”. In Sections S-1 and S-2 we prove detailed balance, irreducibility and aperiodicity of TTMCMC. In Section S-3 we provide the general TTMCMC algorithm for jumping $m$ dimensions, and in Section S-4 we prove detailed balance for this algorithm. The TTMCMC algorithm for jumping more than one dimension at a time when several sets of parameters are related, is detailed in Section S-5. A brief discussion on label switching in mixtures is included in Section S-6, and in Section S-7, an effective method of summarization of the posterior distribution of mixture densities is considered. In Section S-8 we shed light on the important problem of convergence diagnostics in mixture problems, for both known and unknown number of components. Further simulation studies with gamma mixtures with varying data sizes are elaborated in Section S-9. In Section S-10 we include a detailed comparison between additive TTMCMC and random walk RJMCMC with respect to three real data sets, and finally, in Section S-11, we provide detailed comparisons between additive TTMCMC and RJMCMC with respect to the prior structure and the algorithm of Richardson and Green (1997) in the galaxy data context.