ReviewExact posterior computation in non-conjugate Gaussian location-scale parameters models

- In traditional Bayesian statistical analysis, non-conjugate models have no exact posterior distribution.
- In a Gaussian location-scale parameter structure model, the posterior distribution is bidimensional and was only computable by approximate methods.
- The use of special functions allows to obtain a large class of non-conjugate Gaussian models in a exact form.
- The quantities needed to make inferences from the posterior distribution are provided.

In Bayesian analysis the class of conjugate models allows to obtain exact posterior distributions, however this class quite restrictive in the sense that it involves only a few distributions. In fact, most of the practical applications involves non-conjugate models, thus approximate methods, such as the MCMC algorithms, are required. Although these methods can deal with quite complex structures, some practical problems can make their applications quite time demanding, for example, when we use heavy-tailed distributions, convergence may be difficult, also the Metropolis-Hastings algorithm can become very slow, in addition to the extra work inevitably required on choosing efficient candidate generator distributions. In this work, we draw attention to the special functions as a tools for Bayesian computation, we propose an alternative method for obtaining the posterior distribution in Gaussian non-conjugate models in an exact form. We use complex integration methods based on the H-function in order to obtain the posterior distribution and some of its posterior quantities in an explicit computable form. Two examples are provided in order to illustrate the theory.