  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.
