  Max-stable processes provide natural models for the modelling of spatial
extreme values observed at a set of spatial sites. Full likelihood inference
for max-stable data is, however, complicated by the form of the likelihood
function as it contains a sum over all partitions of sites. As such, the number
of terms to sum over grows rapidly with the number of sites and quickly becomes
prohibitively burdensome to compute.
  We propose a variational inference approach to full likelihood inference that
circumvents the problematic sum. To achieve this, we first posit a parametric
family of partition distributions from which partitions can be sampled. Second,
we optimise the parameters of the family in conjunction with the max-stable
model to find the partition distribution best supported by the data, and to
estimate the max-stable model parameters.
  In a simulation study we show that our method enables full likelihood
inference in higher dimensions than previous methods, and is readily applicable
to data sets with a large number of observations. Furthermore, our method can
easily be extended to a Bayesian setting. Code is available at
https://github.com/LPAndersson/MaxStableVI.jl.
