# Algorithmic Bayesian Group Gibbs Selection

## TL;DR

This paper introduces an exact Bayesian group selection algorithm using Gibbs sampling, optimized for various noise models, with applications in genetics and high-dimensional data analysis.

## Contribution

It develops a novel, exact selection engine for Bayesian models that efficiently handles group effects and diverse noise types, with new MCMC steps ensuring rapid convergence.

## Key findings

- Effective group selection in genetic studies.
- Application to high-dimensional phenotype data.
- Fast convergence of the proposed MCMC method.

## Abstract

Bayesian model selection, with precedents in George and McCulloch (1993) and Abramovich et al. (1998), support credibility measures that relate model uncertainty, but computation can be costly when sparse priors are approximate. We design an exact selection engine suitable for Gauss noise, t-distributed noise, and logistic learning, benefiting from data-structures derived from coordinate descent lasso. Gibbs sampler chains are stored in a compressed binary format compatible with Equi-Energy (Kou et al., 2006) tempering. We achieve a grouped-effects selection model, similar to the setting for group lasso, to determine co-entry of coefficients into the model. We derive a functional integrand for group inclusion, and introduce a new MCMC switching step to avoid numerical integration. Theorems show this step has exponential convergence to target distribution. We demonstrate a role for group selection to inform on genetic decomposition in a diallel experiment, and identify potential quantitative trait loci in p > 40K Heterogenous Stock haplotype/phenotype studies.