The regression model for the response variable with bounded domain is discussed. The baseline distribution called the unit Lindley distribution is considered. In the context of regression structure, the logit function is utilized with the unit Lindley model. Then, we have developed the Bayesian unit Lindley regression based on a frequently used prior. Additionally, we also investigate the specific prior for all standardized exploratory variables. The syntax of JAGS for the proposed model is included. In application study, the Bayesian unit Lindley regression is applied to two different datasets where response variables are associated with gasoline yield and risk assessment respectively. Based on the result of estimates and log-likelihood values, it is important to point out that the Bayesian unit-Lindley regression can improve the performance of the classical one.