Analysis of block slice samplers for Bayesian GLMMs and GAMs with linear inequality and shape constraints

TL;DR

This paper introduces a flexible slice sampler Gibbs algorithm for Bayesian GLMMs and GAMs with linear inequality and shape constraints, providing rigorous theoretical guarantees and practical inference tools for constrained models.

Contribution

It develops a novel Bayesian sampling algorithm with proven Markov chain CLT for constrained models, enabling valid finite-sample inference and application to real data.

Findings

Proved uniform ergodicity and CLT for the sampler.

Derived joint bands and multiplicity adjusted inference.

Applied to concussion recovery data with meaningful results.

Abstract

Exponential family models, generalized linear models (GLMs), generalized linear mixed models (GLMMs) and generalized additive models (GAMs) are widely used methods in statistics. However, many scientific applications necessitate constraints be placed on model parameters such as shape and linear inequality constraints. Constrained estimation and inference of parameters remains a pervasive problem in statistics where many methods rely on modifying rigid large sample theory assumptions for inference. We propose a flexible slice sampler Gibbs algorithm for Bayesian GLMMs and GAMs with linear inequality and shape constraints. We prove our posterior samples follow a Markov chain central limit theorem (CLT) by proving uniform ergodicity of our Markov chain and existence of the a moment generating function for our posterior distributions. We use our CLT results to derive joint bands and multiplicity adjusted Bayesian inference for nonparametric functional effects. Our rigorous CLT results address a shortcoming in the literature by obtaining valid estimation and inference on constrained parameters in finite sample settings. Our algorithmic and proof techniques are adaptable to a myriad of important statistical modeling problems. We apply our Bayesian GAM to a real data analysis example involving proportional odds regression for concussion recovery in children with shape constraints and smoothed nonparametric effects. We obtain multiplicity adjusted inference on monotonic nonparametric time effect to elucidate recovery trends in children as a function of time.