  Discrete data are abundant and often arise as counts or rounded data. These
data commonly exhibit complex distributional features such as zero-inflation,
over-/under-dispersion, boundedness, and heaping, which render many parametric
models inadequate. Yet even for parametric regression models, approximations
such as MCMC typically are needed for posterior inference. This paper
introduces a Bayesian modeling and algorithmic framework that enables
semiparametric regression analysis for discrete data with Monte Carlo (not
MCMC) sampling. The proposed approach pairs a nonparametric marginal model with
a latent linear regression model to encourage both flexibility and
interpretability, and delivers posterior consistency even under model
misspecification. For a parametric or large-sample approximation of this model,
we identify a class of conjugate priors with (pseudo) closed-form posteriors.
All posterior and predictive distributions are available analytically or via
direct Monte Carlo sampling. These tools are broadly useful for linear
regression, nonlinear models via basis expansions, and variable selection with
discrete data. Simulation studies demonstrate significant advantages in
computing, prediction, estimation, and selection relative to existing
alternatives. This novel approach is applied successfully to self-reported
mental health data that exhibit zero-inflation, overdispersion, boundedness,
and heaping.
