The block-Poisson estimator for optimally tuned exact subsampling MCMC

Abstract

Speeding up Markov Chain Monte Carlo (MCMC) for datasets with many observations by data subsampling has recently received considerable attention in the literature.The currently available methods are either approximate, highly inefficient or limited to small dimensional models. We propose a pseudo-marginal MCMC method that estimates the likelihood by data subsampling using a block-Poisson estimator. The estimator is a product of Poisson estimators, each based on an independent subset of the observations. The construction allows us to update a subset of the blocks in each MCMC iteration, thereby inducing a controllable correlation between the estimates at the current and proposed draw in the Metropolis-Hastings ratio. This makes it possible to use highly variable likelihood estimators without adversely affecting the sampling efficiency. Poisson estimators are unbiased but not necessarily positive. We therefore follow Lyne et al. (2015) and run the MCMC on the absolute value of the estimator and use an importance sampling correction for occasionally negative likelihood estimates to estimate expectations of any function of the parameters.We provide analytically derived guidelines to select the optimal tuning parameters for the algorithm by minimizing the variance of the importance sampling corrected estimator per unit of computing time. The guidelines are derived under idealized conditions, but are demonstrated to be quite accurate in empirical experiments. The guidelines apply to any pseudo-marginal algorithm if the likelihood is estimated by the block-Poisson estimator, including the class of doubly intractable problems in Lyne et al. (2015). We illustrate the method in a logistic regression example and find dramatic improvements compared to regular MCMC without subsampling and a popular exact subsampling approach recently proposed in the literature.

Speeding Up MCMC by Delayed Acceptance and Data Subsampling

“…Typically a small fraction of observations are included, hence speeding up the execution time significantly compared to MH. However, the augmentation scheme severely affects the mixing of the Firefly algorithm and it has been demonstrated to perform poorly compared to MH and other subsampling approaches (Bardenet et al, 2015;Quiroz et al, 2016Quiroz et al, , 2017. We conclude that delayed acceptance, out of the discussed methods, seems to be the only feasible route to obtain exact simulation via data subsampling.…”

“…Other authors use a subsample of the data in each MCMC iteration to speed up the algorithm, see e.g. Korattikara et al (2014), Bardenet et al (2014), Maclaurin and Adams (2014), Maire et al (2015), Bardenet et al (2015) and Quiroz et al (2016Quiroz et al ( , 2017. Finally, delayed acceptance MCMC has been used to speed up computations (Banterle et al, 2014;Payne and Mallick, 2015).…”

“…This typically requires computations using the full data set, naturally defeating the purpose of subsampling. Quiroz et al (2017) instead suggests to compute a soft lower bound which is defined to be a lower bound with a high probability. Since the estimate can occasionally be negative, the pseudo-marginal in Quiroz et al (2017) instead targets an absolute measure following Lyne et al (2015), who show that the draws can be corrected with an importance sampling type of step to estimate expectations of posterior functions exactly.…”

Flexible Bayesian Nonlinear Model Configuration

“…To reduce the computational complexity of HMC and improve its scalability to large data sets, Welling and Teh (2011) suggested to use stochastic estimates of the gradient of the likelihood. Many recent articles describe the possibility of such sub-sampling combined with MCMC (Quiroz et al, 2019(Quiroz et al, , 2017(Quiroz et al, , 2016Flegal, 2012;Pillai and Smith, 2014), where unbiased likelihood estimates are obtained from subsamples of the whole data set in such a way that ergodicity and the desired limiting properties of the MCMC algorithm are maintained. These methods are not part of the current implementation of BGNLM, but our approach can relatively easily be adapted to allow sub-sampling MCMC techniques.…”

Asymptotically Optimal Exact Minibatch Metropolis-Hastings

“…These minibatch MH methods can be divided into two classes, exact and inexact, depending on whether or not the target distribution π is necessarily preserved. Inexact methods introduce asymptotic bias to the target distribution, trading off correctness for speedups [6,16,20,21,23]. Exact methods either require impractically strong constraints on the target distribution [18,24], limiting their applicability in practice, or they negatively impact efficiency, counteracting the speedups that minibatching aims to provide in the first place [4,10].…”