On exact posterior distributions using H-functions

The use of approximate methods such as MCMC is widespread in Bayesian Analysis. Due to efficient and flexibility of those methods most of the literature in this area has been focused on improving the algorithms, trying to automatise them and addressing problems of convergence. In fact, due to the applications, those methods made Bayesian Inference one of the most prominent modern sciences of the past three decades. Nevertheless, in this work we point to another direction, which allows to obtain analytically the posterior distribution in a great variety of non-conjugate models. We use the theory of special functions in the Bayesian computation context to compute the posterior distribution and its quantities in an exact form. The theory is presented using a general single parameter model based on H-functions, in which we provide a general procedure to obtain posterior distributions which are exact, in the sense that the posterior quantities, such as the moments, the cumulative and the predictive posterior distributions, are explicitly written in a computable form.