conditionally-independent generalized Pareto likelihood, incorporating all spatial dependence through Gaussian process priors on the generalized Pareto likelihood parameters. Spatial dependence has also been achieved through Bayesian Gaussian copula models \[@sang-2010a\] and through a more flexible copula based on a Dirichlet process construction \[@fuentes-2010a\].

We develop a new hierarchical Bayesian model for analyzing max-stable processes. The responses are modeled as independent univariate GEV conditioned on spatial random effects with positive stable random effect distribution. Positive stable random effects have been used to model multivariate extremes with finite dimensions \[@fougeres-2009 [@stephenson-2009]\]. We extend this approach to accommodate data on a continuous spatial domain. We show that the resulting model is max-stable marginally over the random effects, and that a limiting case of this construction provides a finite-dimensional approximation to the well-known Gaussian extreme value process (GEVP) of @smith-1990a, often referred to as the “Smith process.” Lower-dimensional representations have previously been proposed for high-dimensional extremes in various settings \[@pickands-1981 [@Deheuvel-1983; @Schlather-2002; @Ehlert-2008; @wang-2010a; @wang-2010c; @oesting-2011a; @engelke-2011a]\]. Our construction permits analysis of the joint distribution of all observations, and thus can produce straightforward predictions at unobserved locations. Because we use a hierarchical model to represent the spatial max-stable process, a Bayesian implementation is a 