The Triply-Randomized Negative Binomial Beta for Robust Regression and Conjugate Models of Bounded Support Data

Abstract: The beta distribution is the default choice of likelihood in many regression problems with a $[0,1]$-bounded support response despite its sensitivity to outliers, inability to accommodate exact zero observations, and a lack of closed-form conjugate priors. We address these shortcomings by introducing the triply-randomized negative binomial beta distribution, denoted $\mathrm{TNBbeta}(p,\,q,\,\varepsilon)$, parameterized by a median $p$, concentration parameter $q$, and boundary parameter $\varepsilon$ which permits positive density at $0$ and $1$. The TNBbeta arises by randomizing the parameters of a standard beta distribution with three dependent negative binomial random variables, each of whose complete conditional distribution we show is itself negative binomial. Moreover, connecting $p$ and $q$ to Gaussian latent variables with logit link functions yields closed-form updates via Pólya-gamma augmentation. Together, these properties yield simple auxiliary-variable Gibbs samplers for regression models of bounded-support data, which often outperform standard beta regression approaches in terms of effective sample size per second and held-out prediction, especially in the presence of outliers. In a case study of forest canopy cover, we demonstrate that this framework can easily incorporate spatial structure and exact zero observations. Overall, this work substantially expands the class of Bayesian models for $[0,1]$-bounded support data that can be fit efficiently.