An Approach to Incorporate Subsampling Into a Generic Bayesian Hierarchical Model

Abstract

The goal of this article is to provide a way for Bayesian statisticians to incorporate subsampling directly into the Bayesian hierarchical model of their choosing without imposing additional restrictive model assumptions. We are motivated by the fact that the rise of “big data” has created difficulties for statisticians to directly apply their methods to big datasets. We introduce a “data subset model” to the popular “data model, process model, and parameter model” framework used to summarize Bayesian hierarchical models. The hyperparameters of the data subset model are specified constructively in that they are chosen such that the implied size of the subset satisfies predefined computational constraints. Thus, these hyperparameters effectively calibrate the statistical model to the computer itself to obtain predictions/estimations in a prespecified amount of time. Several properties of the data subset model are provided including: propriety, partial sufficiency, and semi-parametric properties. Simulated datasets will be used to assess the consequences of subsampling, and results will be presented across different computers to show the effect of the computer on the statistical analysis. Additionally, we provide a joint analysis of a high-dimensional dataset (roughly 10 gigabytes) consisting of 2018 5-year period estimates from the U.S. Census Bureau’s Public Use Micro-Sample (PUMS).

KEYWORDS:

• Bayesian hierarchical model
• Big data
• Curse of dimensionality
• Gibbs sampler
• Markov chain Monte Carlo
• Non-Gaussian
• Small sample theory

Acknowledgments

I thank the editor, associate editor, and reviewers for their time and feedback on an earlier draft of this article.

Additional information

Funding

This research was partially supported by the U.S. National Science Foundation (NSF) under grant SES-1853099.