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.

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The rapid development of computing power and efficient Markov Chain Monte Carlo (MCMC) simulation algorithms have revolutionized Bayesian statistics, making it a highly practical inference method in applied work. However, MCMC algorithms tend to be computationally demanding, and are particularly slow for large datasets. Data subsampling has recently been suggested as a way to make MCMC methods scalable on massively large data, utilizing efficient sampling schemes and estimators from the survey sampling literature. These developments tend to be unknown by many survey statisticians who traditionally work with non-Bayesian methods, and rarely use MCMC. Our article explains the idea of data subsampling in MCMC by reviewing one strand of work, Subsampling MCMC, a so called pseudo-marginal MCMC approach to speeding up MCMC through data subsampling. The review is written for a survey statistician without previous knowledge of MCMC methods since our aim is to motivate survey sampling experts to contribute to the growing Subsampling MCMC literature.

Structural equation models (SEMs) are commonly used to study the structural relationship between observed variables and latent constructs. Recently, Bayesian fitting procedures for SEMs have received more attention thanks to their potential to facilitate the adoption of more flexible model structures, and variational approximations have been shown to provide fast and accurate inference for Bayesian analysis of SEMs. However, the application of variational approximations is currently limited to very simple, elemental SEMs. We develop mean-field variational Bayes algorithms for two SEM formulations for data that present non-Gaussian features such as skewness and multimodality. The proposed models exploit the use of mixtures of Gaussians, include covariates for the analysis of latent traits and consider missing data. We also examine two variational information criteria for model selection that are straightforward to compute in our variational inference framework. The performance of the MFVB algorithms and information criteria is investigated in a simulated data study and a real data application.