MCMC for Imbalanced Categorical Data
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.
Duke Scholars
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- Statistics & Probability
- 4905 Statistics
- 3802 Econometrics
- 1603 Demography
- 1403 Econometrics
- 0104 Statistics
Citation
Published In
DOI
EISSN
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- Statistics & Probability
- 4905 Statistics
- 3802 Econometrics
- 1603 Demography
- 1403 Econometrics
- 0104 Statistics