Exact Bayesian modeling for bivariate Poisson data and extensions |
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Authors: | Dimitris Karlis Panagiotis Tsiamyrtzis |
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Institution: | (1) Department of Statistics, Athens University of Economics and Business, 76 Patission Str., 10434 Athens, Greece |
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Abstract: | Bivariate count data arise in several different disciplines (epidemiology, marketing, sports statistics just to name a few)
and the bivariate Poisson distribution being a generalization of the Poisson distribution plays an important role in modelling
such data. In the present paper we present a Bayesian estimation approach for the parameters of the bivariate Poisson model
and provide the posterior distributions in closed forms. It is shown that the joint posterior distributions are finite mixtures
of conditionally independent gamma distributions for which their full form can be easily deduced by a recursively updating
scheme. Thus, the need of applying computationally demanding MCMC schemes for Bayesian inference in such models will be removed,
since direct sampling from the posterior will become available, even in cases where the posterior distribution of functions
of the parameters is not available in closed form. In addition, we define a class of prior distributions that possess an interesting
conjugacy property which extends the typical notion of conjugacy, in the sense that both prior and posteriors belong to the
same family of finite mixture models but with different number of components. Extension to certain other models including
multivariate models or models with other marginal distributions are discussed. |
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Keywords: | Bayesian sequentially updated mixtures Conjugacy Direct sampling Gamma mixtures |
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