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.

翻译：最大稳定过程为一组空间站点观测的空间极端值建模提供了自然模型。但是,由于最大稳定数据的概率函数包含对各站点所有分区的总和,因此其可能功能的形式复杂。因此,要加起来的条件数量随着地点数目的迅速增长而迅速增长,并很快成为令人难以承受的计算负担。我们建议了一种变式推论方法,以完全有可能地推论,绕过问题的总和。为了实现这一目标,我们首先假设分区分布分布的参数组,从中取样。第二,我们优化家庭参数与最大稳定模型的参数,以找到数据最支持的分区分布,并估计最大稳定模型参数。在一项模拟研究中,我们表明,我们的方法能够使比以往方法具有更大范围的推论,并且很容易适用于大量观测的数据集。此外,我们的方法可以很容易扩展到巴伊西亚的设置。代码可在 https://github.com/LPAndersson/MaxstableVI上查阅。