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.

Regression analysis is done, traditionally, considering homogeneity and normality assumption for the response variable distribution. Whereas in many applications, observations indicate to a heterogeneous structure containing some sub-populations with skew-symmetric structure either due to heterogeneity, multimodality or skewness of the population or a combination of them. In this situations, one can use a mixture of skew-symmetric distributions to model the population. In this paper we considered the Bayesian approach of regression analysis under the assumption of heterogeneity of population and a skew-symmetric distribution for sub-populations, by using a mixture of skew normal distributions. We used a simulation study and a real world example to assess the proposed Bayesian methodology and to compare it with frequentist approach.