Variational inference on a Bayesian adaptive lasso Tobit quantile regression model

The Tobit quantile regression model is a useful tool for quantifying the relationship between response variables with limited values and the explanatory variables. Under the Bayesian framework, the Tobit quantile regression model is often simulated by an asymmetric Laplacian distribution (ALD), which can be reformulated as a hierarchical structure model. An adaptive lasso prior is used to address the selection of active explanatory variables. A mean-field variational family is adopted, where the variables are assumed to be mutually independent with each being governed by a different factor in the variational density. A coordinate ascent variational inference (CAVI) algorithm is developed to iteratively optimize each factor, and the evidence lower bound (ELBO) is obtained. Parameter estimation and variable selection are simultaneously produced by the optimal variational density. Several simulation studies and an example are presented to illustrate the proposed methodologies.