MCMC for Imbalanced Categorical Data

TL;DR

This paper analyzes the computational efficiency of MCMC algorithms for highly imbalanced categorical data, revealing that adaptive Metropolis outperforms data augmentation in large samples due to differences in convergence behavior.

Contribution

It provides theoretical insights into the computational complexity of MCMC methods in imbalanced data settings, explaining why some algorithms perform poorly as data size grows.

Findings

Metropolis has logarithmic complexity in sample size

Data augmentation has polynomial complexity in sample size

Discrepancy in concentration rates causes poor performance of data augmentation

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 MCMC approaches. This article was motivated by an application to quantitative advertising in which we encountered extremely poor computational performance for common data augmentation MCMC algorithms but obtained excellent performance for adaptive Metropolis. To obtain a deeper understanding of this behavior, we give strong theory results on computational complexity in an infinitely imbalanced asymptotic 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 poor performance of data augmentation is a discrepancy between the rates at which the target density and MCMC step sizes concentrate. In general, MCMC algorithms that have a similar discrepancy will fail in large samples - a result with substantial practical impact.