Hierarchical Bayesian LASSO for a negative binomial regression

Numerous researches have been carried out to explain the relationship between the count data y and numbers of covariates x through a generalized linear model (GLM). This paper proposes a hierarchical Bayesian least absolute shrinkage and selection operator (LASSO) solution using six different prior models to the negative binomial regression. Latent variables Z have been introduced to simplify the GLM to a standard linear regression model. The proposed models regard two conjugate zero-mean Normal priors for the regression parameters and three independent priors for the variance: the Exponential, Inverse-Gamma and Scaled Inverse-chi(2) distributions. Different types of priors result in different amounts of shrinkage. A Metropolis-Hastings-within-Gibbs algorithm is used to compute the posterior distribution of the parameters of interest through a data augmentation process. Based on the posterior samples, an original double likelihood ratio test statistic have been proposed to choose the most relevant covariates and shrink the insignificant coefficients to zero. Numerical experiments on a real-life data set prove that Bayesian LASSO methods achieved significantly better predictive accuracy and robustness than the classical maximum likelihood estimation and the standard Bayesian inference.