Bayesian inference using Gibbs sampling for Window version (WinBUGS), software for Bayesian analysis using MCMC method and Gibbs sampler

This paper presents two types of symmetric scale mixture probability distributions which include the normal, Student t, Pearson Type VII, variance gamma, exponential power, uniform power and generalized t (GT) distributions. Expressing a symmetric distribution into a scale mixture form enables efficient Bayesian Markov chain Monte Carlo (MCMC) algorithms in the implementation of complicated statistical models. Moreover, the mixing parameters, a by-product of the scale mixture representation, can be used to identify possible outliers. This paper also proposes a uniform scale mixture representation for the GT density, and demonstrates how this density representation alleviates the computational burden of the Gibbs sampler.