  We propose a simple yet powerful framework for modeling integer-valued data,
such as counts, scores, and rounded data. The data-generating process is
defined by Simultaneously Transforming and Rounding (STAR) a continuous-valued
process, which produces a flexible family of integer-valued distributions
capable of modeling zero-inflation, bounded or censored data, and over- or
underdispersion. The transformation is modeled as unknown for greater
distributional flexibility, while the rounding operation ensures a coherent
integer-valued data-generating process. An efficient MCMC algorithm is
developed for posterior inference and provides a mechanism for adaptation of
successful Bayesian models and algorithms for continuous data to the
integer-valued data setting. Using the STAR framework, we design a new Bayesian
Additive Regression Tree (BART) model for integer-valued data, which
demonstrates impressive predictive distribution accuracy for both synthetic
data and a large healthcare utilization dataset. For interpretable
regression-based inference, we develop a STAR additive model, which offers
greater flexibility and scalability than existing integer-valued models. The
STAR additive model is applied to study the recent decline in Amazon river
dolphins.
