Bayesian Analysis Using a Constrained Mixture of Normal-Inverse-Gamma Models

2026-06-22

AI Summary

This study addresses the severe computational bottleneck in Gaussian mixture regression models under Gibbs sampling, which arises from the need to update discrete latent variables whose number grows with sample size. The authors propose an efficient Bayesian inference approach that leverages a Normal-Inverse-Gamma prior to decompose the posterior into conditionally conjugate parameter distributions amenable to direct sampling and a marginal posterior over mixture labels. Innovatively integrating ideas from Bayesian selective inference, the method restricts the label space to a subset generated by an initial estimate, dramatically improving sampling efficiency. Extensive simulations demonstrate superior performance over conventional MCMC algorithms, and real-world applicability is confirmed through analysis of CDC birth data, highlighting both the method’s effectiveness and practical utility.

Abstract

Gaussian mixtures of regressions are commonly implemented via a Gibbs sampler. This Markov chain Monte Carlo (MCMC) algorithm can be computationally burdensome because of the need to update discrete-valued latent component allocation parameters whose dimension increases as the sample size increases. In this article, we propose applying the method of composition to a Gaussian finite mixture model with a Normal-Inverse-Gamma (NIG) prior which allows one to write the posterior distribution as the product of conditional distributions. Namely, the conditional distribution of parameters given the data and mixture labels, times the marginal posterior of the mixture labels. The conditional distribution of parameters given the data and mixture labels, can be sampled from directly, instead of using MCMC. The expression of the marginal posterior of the mixture labels is known up to a proportionality constant and we adapt existing approaches in Bayesian selective inference to constrain the space of component labels to those arising from preliminary estimators, which alleviates a commonly encountered bottleneck. In simulation studies, we consider several settings and compare several versions of our constrained mixture of NIG models to two different MCMC-based strategies and demonstrate their use on natality data from the CDC.

Problem

• Gaussian mixture regression
• Gibbs sampler
• computational burden
• latent allocation parameters
• Bayesian inference

Innovation

• Constrained mixture model
• Normal-Inverse-Gamma prior
• Method of composition
• Bayesian selective inference
• Direct posterior sampling