Bayesian Mixture Models for Histograms: with Applications to Large Datasets

2026-06-22

AI Summary

This study addresses the challenge of inferring underlying population distributions from aggregated data available only in the form of histograms or frequency tables. The authors propose a nonparametric Bayesian inference method based on mixture models, employing reversible-jump Markov chain Monte Carlo to fit Gaussian mixtures with either a finite or countably infinite number of components. This work represents the first systematic application of a nonparametric Bayesian framework to histogram data analysis. Furthermore, by leveraging Dirichlet processes, the approach jointly models multiple histograms, enabling information sharing across groups and providing posterior probabilities to quantify homogeneity among them. Empirical evaluations demonstrate that the method effectively reconstructs complex distributions from large-scale aggregated data and offers principled clustering and homogeneity assessment.

Abstract

In many real-world scenarios, especially those involving privacy constraints or data summarization, data are available only in aggregated forms, such as histograms or frequency tables. This work introduces a novel Bayesian method for inferring the underlying population distribution by fitting a mixture model to binned data. While we focus on mixtures of normal distributions, the framework is flexible and can be extended to other distributional families. We place a prior distribution on the number of mixture components, accommodating both finite and countably infinite mixtures, and perform inference using reversible jump MCMC. The proposed approach demonstrates strong performance on large-scale data, showcasing the potential of nonparametric Bayesian modeling in practical applications. Furthermore, we extend the method to model multiple histograms simultaneously and cluster them using the Dirichlet process. This enables information sharing across populations and provides a principled posterior probability to assess homogeneity between groups. Some theoretical results supporting the performance of our proposed methodology are also discussed.

Problem

• Bayesian mixture models
• histograms
• binned data
• population distribution inference
• large datasets

Innovation

• Bayesian mixture models
• binned data
• reversible jump MCMC
• Dirichlet process
• nonparametric Bayesian