In this  paper  the  regression analysis with finite mixture bivariate poisson response variable is investigated from the Bayesian point of view. It is shown that  the posterior distribution can not be written in a closed form due to the  complexity of the likelihood function of bivariate Poisson distribution. Hence, the full conditional posterior distributions of the parameters are computed and the Gibbs algorithm is used to sampling from posterior distributions. A simulation study is performed in order to assess the proposed Bayesian model and its efficiency in estimation of the parameters is compared with their frequentist counterparts. Also, a real example presented to illustrate and assess the proposed Bayesian model. The results indicate to the more efficiency of the  estimators resulted from Bayesian  approach than estimators of frequentist approach at least for small sample sizes.