Penalising the complexity of extensions of the Gaussian distribution

The Gaussian distribution has ever been the most popular and usable device in the field of statistics. Even in the context of penalised complexity (PC) priors, the normal density has a particular meaning, especially because we can consider it as a base model which could be extended both in terms of tail thickness and skewness. We derive the numerical PC prior for the shape parameter of the skew-normal density and the analytical PC prior for the degrees of freedom of the t-distribution. We also perform an approximation of the Kullback-Leibler divergence (KLD) in the the skew-normal model