Bayesian Methods for Non-stationary Extreme Value Analysis

Abstract

Non-stationary models for extremes have attracted significant attention in recent years. These models require adapted estimation methods. Bayesian inference offers an attractive framework to estimate non-stationary models and, importantly, to quantify estimation and predictive uncertainties.

This chapter therefore focuses on the application of Bayesian inference to non-stationary extreme models. It is organized as a step-by-step building of non-stationary models of increasing generality. The principles of Bayesian inference are introduced using the simple case of a univariate and stationary distribution. The construction of at-site non-stationary models, using regression functions linking parameter values with time-varying covariates, is then presented. The difficulty of identifying non-stationary components based on the sole use of at-site data is also discussed, and motivates the construction of regional non-stationary models. Such models are based on the concept of “regional parameters”, i.e. parameters being assumed identical for all sites within a homogeneous region. The inference of regional models poses an additional difficulty compared to the at-site approach: the existence of spatial dependences makes the derivation of the inference equations challenging. A practical solution, based on the use of spatial copulas, is briefly presented. Lastly, a generalization of the “regional parameter” paradigm, based on Bayesian hierarchical modeling, is discussed.

Appendix

3.1.1 The Chib Method for Computing Marginal Likelihoods

The marginal likelihood is the normalizing constant of the posterior distribution. For a given model M (omitted from the notation for simplicity), it can therefore be written as follows:

$$ p(\user2{y}) = \frac{{p(\user2{y}|\user2{\theta })p(\user2{\theta })}}{{p(\user2{\theta }|\user2{y})}} = \frac{{f(\user2{\theta })}}{{p(\user2{\theta }|\user2{y})}} $$

(3.51)

In this equation, the numerator is the unnormalized posterior pdf and can therefore be computed for any θ. The difficulty is to compute the denominator, i.e. the normalized posterior pdf $p(\user2{\theta }|\user2{y})$.

Chib’s approach (1995) to this problem is first based on the observation that relation (3.51) holds for any θ value. For a given value ${{\user2{\theta }}^{*}}$, it is then possible to decompose the normalized posterior pdf $p({{\user2{\theta }}^{*}}|\user2{y})$ evaluated at ${{\user2{\theta }}^{*}}$ as follows:

$$ \begin{aligned}[b] p({{\user2{\theta }}^{*}}|\user2{y}) & = p(\theta_1^{*}|\user2{y})p(\theta_2^{*}|\theta_1^{*},\user2{y}) \times ... \times p(\theta_k^{*}|\theta_1^{*},...,\theta_{{k - 1}}^{*},\user2{y}) \times ... \\ & \quad \times p(\theta_{{{{N}_D}}}^{*}|\theta_1^{*},...,\theta_{{{{N}_D} - 1}}^{*},\user2{y})\end{aligned} $$

(3.52)

Equation (3.52) decomposes the computation of a N _D -dimensional pdf into the multiplication of N _D one-dimensional pdfs. Each term $p( \theta_k^{*}|\theta_1^{*},...,\theta_{{k - 1}}^{*},\user2{y})$ is the first marginal distribution of the distribution $p( {{\theta }_k},{{\theta }_{{k + 1}}}...,{{\theta }_{{{{N}_D}}}}|\theta_1^{*},...,\theta_{{k - 1}}^{*},\user2{y})$, evaluated at $\theta_k^{*}$. Consequently, Chib’s proposal is to perform additional MCMC sampling from the conditional distributions $p( {{\theta }_k},{{\theta }_{{k + 1}}}...,{{\theta }_{{{{N}_D}}}}|\theta_1^{*},...,\theta_{{k - 1}}^{*},\user2{y})$ for k = 2:N _D , and to use the first marginal sample to compute each term $p( \theta_k^{*}|\theta_1^{*},...,\theta_{{k - 1}}^{*},\user2{y})$. Since the latter is a one-dimensional distribution, estimating its normalized pdf based on MCMC samples poses no difficulty, either using a kernel density estimate or one-dimensional numerical integration.

Although Chib’s approach is computationally demanding in high-dimensional problems, it remains one of the most stable approximations of the marginal likelihood (Bos 2002). Note that ${{\user2{\theta }}^{*}}$ should preferably be chosen in a high-density area of the posterior distribution (e.g. the posterior mode) to improve the efficiency of the approximation.