# Consistency of Bayesian Inference for Multivariate Max-Stable Distributions

## TL;DR

This paper proves the consistency of a Bayesian semiparametric method for estimating multivariate max-stable distributions, crucial for predicting extreme events in risk analysis, even when data are only approximately max-stable.

## Contribution

It establishes the first rigorous consistency results for Bayesian inference in multivariate max-stable models, including near-model scenarios.

## Key findings

- Posterior distributions are consistent for well-specified models.
- Consistency extends to data near max-stable distributions.
- Results apply to models with various tail behaviors.

## Abstract

Predicting extreme events is important in many applications in risk analysis. The extreme-value theory suggests modelling extremes by max-stable distributions. The Bayesian approach provides a natural framework for statistical prediction. Although various Bayesian inferential procedures have been proposed in the literature of univariate extremes and some for multivariate extremes, the study of their asymptotic properties has been left largely untouched. In this paper we focus on a semiparatric Bayesian method for estimating max-stable distributions in arbitrary dimension. We establish consistency of the pertaining posterior distributions for fairly general, well-specified max-stable models, whose margins can be short-, light- or heavy-tailed. We then extend our consistency results to the case where the data come from a distribution lying in a neighbourhood of a max-stable one, which represents the most realistic inferential setting.