A Bootstrap Metropolis–Hastings Algorithm for Bayesian Analysis of Big Data

Markov chain Monte Carlo (MCMC) methods have proven to be a very powerful tool for analyzing data of complex structures. However, their computer-intensive nature, which typically require a large number of iterations and a complete scan of the full dataset for each iteration, precludes their use for big data analysis. In this article, we propose the so-called bootstrap Metropolis–Hastings (BMH) algorithm that provides a general framework for how to tame powerful MCMC methods to be used for big data analysis, that is, to replace the full data log-likelihood by a Monte Carlo average of the log-likelihoods that are calculated in parallel from multiple bootstrap samples. The BMH algorithm possesses an embarrassingly parallel structure and avoids repeated scans of the full dataset in iterations, and is thus feasible for big data problems. Compared to the popular divide-and-combine method, BMH can be generally more efficient as it can asymptotically integrate the whole data information into a single simulation run. The BMH algorithm is very flexible. Like the Metropolis–Hastings algorithm, it can serve as a basic building block for developing advanced MCMC algorithms that are feasible for big data problems. This is illustrated in the article by the tempering BMH algorithm, which can be viewed as a combination of parallel tempering and the BMH algorithm. BMH can also be used for model selection and optimization by combining with reversible jump MCMC and simulated annealing, respectively. Supplementary materials for this article are available online.