Global-Local Mixtures

We show how to generate global-local mixtures using the Cauchy-Schlomilch and the Liouville integral transformation identities. This leads to simpler characterizations of well-known normal-scale mixture distributions such as the Laplace or Lasso, the logit and the quantile. We also show how they lead to new probability distributions as global-local mixtures of appropriate baseline densities. Finally, we conclude with a conjecture concerning the Bayesian bridge and uniform correlation mixture of a bivariate normal density with unit variances.