MCMC for Imbalanced Categorical Data

ABSTRACT

Many modern applications collect highly imbalanced categorical data, with some categories relatively rare. Bayesian hierarchical models combat data sparsity by borrowing information, while also quantifying uncertainty. However, posterior computation presents a fundamental barrier to routine use; a single class of algorithms does not work well in all settings and practitioners waste time trying different types of Markov chain Monte Carlo (MCMC) approaches. This article was motivated by an application to quantitative advertising in which we encountered extremely poor computational performance for data augmentation MCMC algorithms but obtained excellent performance for adaptive Metropolis. To obtain a deeper understanding of this behavior, we derive theoretical results on the computational complexity of commonly used data augmentation algorithms and the Random Walk Metropolis algorithm for highly imbalanced binary data. In this regime, our results show computational complexity of Metropolis is logarithmic in sample size, while data augmentation is polynomial in sample size. The root cause of this poor performance of data augmentation is a discrepancy between the rates at which the target density and MCMC step sizes concentrate. Our methods also show that MCMC algorithms that exhibit a similar discrepancy will fail in large samples—a result with substantial practical impact. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian
• Data augmentation
• Gibbs sampling
• Large sample
• Markov chain Monte Carlo
• Mixing

Funding

J. Johndrow and D. Dunson acknowledge support from the United States National Science Foundation award number 1546130 and MaxPoint Interactive. A. Smith acknowledges support from the Natural Sciences and Engineering Research Council of Canada.