We propose Bayesian inference for bivariate Poisson models that generalizes the existing approaches in two important directions. Firstly we propose exact inference contrary to the MCMC approaches existing in the literature and secondly we use a prior distribution that allows for dependencies among the parameters of interest. Our prior is in fact a mixture of priors and the resulting posterior generalizes the idea of conjugacy in the sense that it is again a mixture of the same family but with more components. Computational details and a real data illustration are provided. Extensions of our approach to certain other models is discussed.
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