Markov chain Monte Carlo based on deterministic transformations

Abstract

In this article we propose a novel MCMC method based on deterministic transformations T : X ×D → X where X is the state-space and D is some set which may or may not be a subset of X . We refer to our new methodology as Transformation-based Markov chain Monte Carlo (TMCMC). One of the remarkable advantages of our proposal is that even if the underlying target distribution is very high-dimensional, deterministic transformation of a one-dimensional random variable is sufficient to generate an appropriate Markov chain that is guaranteed to converge to the high-dimensional target distribution. Apart from clearly leading to massive computational savings, this idea of deterministically transforming a single random variable very generally leads to excellent acceptance rates, even though all the random variables associated with the high-dimensional target distribution are updated in a single block. Since it is well-known that joint updating of many random variables using Metropolis-Hastings (MH) algorithm generally leads to poor acceptance rates, TMCMC, in this regard, seems to provide a significant advance. We validate our proposal theoretically, establishing the convergence properties. Furthermore, we show that TMCMC can be very effectively adopted for simulating from doubly intractable distributions.We show that TMCMC includes hybrid Monte Carlo (HMC) as a special case. We also contrast TMCMC with the generalized Gibbs and Metropolis methods of Liu and Yu (1999), Liu and Sabatti (2000) and Kou et al. (2005), pointing out that even though the latter also use transformations, their goal is to seek improvement of the standard Gibbs and Metropolis Hastings algorithms by adding a transformation-based step, while TMCMC is an altogether new and general methodology for simulating from intractable, particularly, high-dimensional distributions.TMCMC is compared with MH using the well-known Challenger data, demonstrating the effectiveness of of the former in the case of highly correlated variables. Moreover, we apply our methodology to a challenging posterior simulation problem associated with the geostatistical model of Diggle * Corresponding et al. (1998), updating 160 unknown parameters jointly, using a deterministic transformation of a one-dimensional random variable. Remarkable computational savings as well as good convergence properties and acceptance rates are the results.

A Bayesian semiparametric approach to learning about gene–gene interactions in case-control studies

“…, d. Then assuming that the target distribution is proportional to π, the new move T (2014) described another TMCMC algorithm that uses the additive transformation for some co-ordinates of x and the multiplicative transformation for the remaining co-ordinates. Dutta & Bhattacharya (2014) refer to this as additive-multiplicative TMCMC. Let the target density π be supported on R d .…”

Evolutionary Sequential Monte Carlo Samplers for Change-Point Models

“…Note that Equation (21) is slightly different from the standard walk move of [19] in the sense that the random parameter δ can be greater than one and also because only one realization from Z W is generated in order to update an entire new vector x j t . The latter modification is motivated by the success of the novel MCMC algorithm based on deterministic proposals (see the Transformation-based MCMC of [26]) and by the DREAM update which also exhibits one (fixed) parameter F(δ, d) to propose the entire new vector.…”

Cross‐validation based assessment of a new Bayesian palaeoclimate model

“…For updating the one‐dimensional variable x , random walk Metropolis with appoximately optimized scaling constant will be used. In fact, Dutta and Bhattacharya () show that a TMCMC step for updating one‐dimensional parameter coincides with a Metropolis–Hasting step; in this case, the additive TMCMC step is equivalent to a random walk Metropolis step. All the other variables will be updated in the way described in Section S‐1.…”

“…Apart from the very much improved results, our model and methods facilitate very fast and efficient computation, which is crucial for palaeoclimate reconstruction where the data sets tend to be (at least moderately) large. For the cross‐validation purpose, we combine the Importance Resampling MCMC (IRMCMC) methodology of Bhattacharya and Haslett () with the recently developed Transformation‐based MCMC (TMCMC) (Dutta and Bhattacharya ()) to further improve computational efficiency. A brief overview of TMCMC is provided in Section 3.1; here we just note that TMCMC allows updating high‐dimensional parameter vectors using simple deterministic transformations of one‐dimensional random variables having arbitrary distributions on some relevant support.…”