  The tail of a bivariate distribution function in the domain of attraction of
a bivariate extreme-value distribution may be approximated by the one of its
extreme-value attractor. The extreme-value attractor has margins that belong to
a three-parameter family and a dependence structure which is characterised by a
spectral measure, that is a probability measure on the unit interval with mean
equal to one half. As an alternative to parametric modelling of the spectral
measure, we propose an infinite-dimensional model which is at the same time
manageable and still dense within the class of spectral measures. Inference is
done in a Bayesian framework, using the censored-likelihood approach. In
particular, we construct a prior distribution on the class of spectral measures
and develop a trans-dimensional Markov chain Monte Carlo algorithm for
numerical computations. The method provides a bivariate predictive density
which can be used for predicting the extreme outcomes of the bivariate
distribution. In a practical perspective, this is useful for computing rare
event probabilities and extreme conditional quantiles. The methodology is
validated by simulations and applied to a data-set of Danish fire insurance
claims.
