A general construction for parallelizing Metropolis−Hastings algorithms

Abstract

Markov chain Monte Carlo methods (MCMC) are essential tools for solving many modern-day statistical and computational problems; however, a major limitation is the inherently sequential nature of these algorithms. In this paper, we propose a natural generalization of the Metropolis−Hastings algorithm that allows for parallelizing a single chain using existing MCMC methods. We do so by proposing multiple points in parallel, then constructing and sampling from a finite-state Markov chain on the proposed points such that the overall procedure has the correct target density as its stationary distribution. Our approach is generally applicable and straightforward to implement. We demonstrate how this construction may be used to greatly increase the computational speed and statistical efficiency of a variety of existing MCMC methods, including Metropolis-Adjusted Langevin Algorithms and Adaptive MCMC. Furthermore, we show how it allows for a principled way of using every integration step within Hamiltonian Monte Carlo methods; our approach increases robustness to the choice of algorithmic parameters and results in increased accuracy of Monte Carlo estimates with little extra computational cost. S ince its introduction in the 1970s, the Metropolis−Hastings algorithm has revolutionized computational statistics (1). The ability to draw samples from an arbitrary probability distribution, πðXÞ, known only up to a constant, by constructing a Markov chain that converges to the correct stationary distribution has enabled the practical application of Bayesian inference for modeling a huge variety of scientific phenomena, and has resulted in Metropolis−Hastings being noted as one of the top 10 most important algorithms from the 20th century (2). Despite regular increases in available computing power, Markov chain Monte Carlo (MCMC) algorithms can still be computationally very expensive; many thousands of iterations may be necessary to obtain low-variance estimates of the required quantities with an oftentimes complex statistical model being evaluated for each set of proposed parameters. Furthermore, many Metropolis−Hastings algorithms are severely limited by their inherently sequential nature.Many approaches have been proposed for improving the statistical efficiency of MCMC, and although such algorithms are guaranteed to converge asymptotically to the stationary distribution, their performance over a finite number of iterations can vary hugely. Much research effort has therefore focused on developing transition kernels that enable moves to be proposed far from the current point and subsequently accepted with high probability, taking into account, for example, the correlation structure of the parameter space (3, 4), or by using Hamiltonian dynamics (5) or diffusion processes (6). A recent investigation into proposal kernels suggests that more exotic distributions, such as the Bactrian kernel, might also be used to increase the statistical efficiency of MCMC algorithms (7). The efficient exploration of high-dimensional and multimo...

A review of multiple try MCMC algorithms for signal processing

“…. , θ (N ) (a similar approach was proposed in [20]). Each resampled candidate is then tested as possible 180 future state of one chain.…”

“…In this work, we provide an exhaustive review of more sophisticated MCMC methods that, at each iteration, consider different candidates as possible new state of the chain. More specifically, at each iteration different samples are compared by certain weights and then one of them is 20 selected as possible future state. The main advantage of these algorithms is that they foster the exploration of a larger portion of the sample space, decreasing the correlation among the states of the generated chain.…”

Statistical methods and computing for big data

“…General strategies for parallel MCMC such as multiple-proposal MH algorithm (Calderhead, 2014) and population MCMC (Song, Wu and Liang, 2014) mostly require full data at each node.…”

Pitching single-focus confocal data analysis one photon at a time with Bayesian nonparametrics

“…where α = α N + b n and β = β N −1 N + 1 − b n . Once the weights q n or all biomolecules are updated from the above procedure, we update the loads b n by sampling from the corresponding conditional distribution P ({b n } n |D, µ part , {q n , x n , y n , z n } n , ∆t) using a Methropolis-Hasting algorithm [17,21]. For this, we use a proposal distribution with the form b n new ∼ Bernoulli(q n ).…”