Posterior propriety in Bayesian extreme value analyses using reference priors

Northrop, Paul J.; Attalides, Nicolas

doi:10.5705/ss.2014.034

Cited by 30 publications

(30 citation statements)

References 36 publications

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“…A priori all GP hyperparameters are independent. For the shape parameter

ξ

of the GEV distribution, either a diffuse prior or a weakly informative normal prior is possible (Northrop and Attalides, ). We will discuss the posterior consistency of our model under these priors in next section.…”

Section: Gp‐gev Binary Regression Modelmentioning

confidence: 99%

Flexible Link Functions in Nonparametric Binary Regression with Gaussian Process Priors

Wang

Lin

et al. 2015

Biometrics

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Summary In many scientific fields, it is a common practice to collect a sequence of 0-1 binary responses from a subject across time, space, or a collection of covariates. Researchers are interested in finding out how the expected binary outcome is related to covariates, and aim at better prediction in the future 0-1 outcomes. Gaussian processes have been widely used to model nonlinear systems; in particular to model the latent structure in a binary regression model allowing nonlinear functional relationship between covariates and the expectation of binary outcomes. A critical issue in modeling binary response data is the appropriate choice of link functions. Commonly adopted link functions such as probit or logit links have fixed skewness and lack the flexibility to allow the data to determine the degree of the skewness. To address this limitation, we propose a flexible binary regression model which combines a generalized extreme value link function with a Gaussian process prior on the latent structure. Bayesian computation is employed in model estimation. Posterior consistency of the resulting posterior distribution is demonstrated. The flexibility and gains of the proposed model are illustrated through detailed simulation studies and two real data examples. Empirical results show that the proposed model outperforms a set of alternative models, which only have either a Gaussian process prior on the latent regression function or a Dirichlet prior on the link function.

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“…A priori all GP hyperparameters are independent. For the shape parameter

ξ

Section: Gp‐gev Binary Regression Modelmentioning

confidence: 99%

Flexible Link Functions in Nonparametric Binary Regression with Gaussian Process Priors

Wang

Lin

et al. 2015

Biometrics

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“…As stated in Section 2.2, suppose that the excess distribution F u of cluster maxima is a GPD with scale σ > 0 and shape ξ ∈ R. Further assume an improper prior for these parameters given, for all σ > 0 and ξ ∈ R, by f .σ,ξ/ .σ, ξ/ ∝ 1=σ. Note that this prior yields a proper posterior as long as the sample size is greater than 2 (Northrop and Attalides, 2016), which is the case here. Bayesian estimates and associated 95% credible intervals for the parameters are then given bŷ σ = 8:6086 ∈ .7:1258, 10:2472/, ξ = 0:0630 ∈ .−0:0464, 0:2056/:…”

Section: Bayesian Fitting Of the Distribution Of Cluster Maximamentioning

confidence: 92%

“…Further assume an improper prior for these parameters given, for all σ >0 and

ξ \in R

, by f ( σ , ξ ) ( σ , ξ )∝1/ σ . Note that this prior yields a proper posterior as long as the sample size is greater than 2 (Northrop and Attalides, ), which is the case here. Bayesian estimates and associated 95% credible intervals for the parameters are then given by

\begin{matrix} \overset{false^}{σ} = 8.6086 \in false(7.1258, 10.2472 false), \\ \overset{false^}{ξ} = 0.0630 \in false(- 0.0464, 0.2056 false) . \end{matrix}

…”

Section: Application To the Burlington Precipitation Datamentioning

confidence: 99%

Modelling Extreme Rain Accumulation with an Application to the 2011 Lake Champlain Flood

Jalbert

Murphy

Genest

et al. 2019

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary A simple strategy is proposed to model total accumulation in non‐overlapping clusters of extreme values from a stationary series of daily precipitation. Assuming that each cluster contains at least one value above a high threshold, the cluster sum S is expressed as the ratio S=M/P of the cluster maximum M and a random scaling factor P ∈ (0,1]. The joint distribution for the pair (M,P) is then specified by coupling marginal distributions for M and P with a copula. Although the excess distribution of M is well approximated by a generalized Pareto distribution, it is argued that, conditionally on P<1, a scaled beta distribution may already be sufficiently rich to capture the behaviour of P. An appropriate copula for the pair (M,P) can also be selected by standard rank‐based techniques. This approach is used to analyse rainfall data from Burlington, Vermont, and to estimate the return period of the spring 2011 precipitation accumulation which was a key factor in that year's devastating flood in the Richelieu Valley Basin in Québec, Canada.

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“…Castellanos and Cabras () showed that the Jeffreys prior yields a proper posterior for

n_{u} ⩾ 1

and Northrop and Attalides () showed that under the flat prior a sufficient condition for posterior propriety is

n_{u} ⩾ 3

. Northrop and Attalides () also showed that, for any sample size, if, and only if, ξ is bounded below a priori , the MDI prior, and the generalized MDI prior, yields a proper posterior. The way in which MDI priors are constructed (Zellner (), section ) means that the functional form of prior (11) is invariant to the particular lower bound that is chosen.…”

Section: Single‐threshold Selectionmentioning

confidence: 99%

Cross-Validatory Extreme Value Threshold Selection and Uncertainty with Application to Ocean Storm Severity

Northrop

Attalides

Jonathan

2016

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary Design conditions for marine structures are typically informed by threshold‐based extreme value analyses of oceanographic variables, in which excesses of a high threshold are modelled by a generalized Pareto distribution. Too low a threshold leads to bias from model misspecification, and raising the threshold increases the variance of estimators: a bias–variance trade‐off. Many existing threshold selection methods do not address this trade‐off directly but rather aim to select the lowest threshold above which the generalized Pareto model is judged to hold approximately. In the paper Bayesian cross‐validation is used to address the trade‐off by comparing thresholds based on predictive ability at extreme levels. Extremal inferences can be sensitive to the choice of a single threshold. We use Bayesian model averaging to combine inferences from many thresholds, thereby reducing sensitivity to the choice of a single threshold. The methodology is applied to significant wave height data sets from the northern North Sea and the Gulf of Mexico.

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Posterior propriety in Bayesian extreme value analyses using reference priors

Cited by 30 publications

References 36 publications

Flexible Link Functions in Nonparametric Binary Regression with Gaussian Process Priors

Flexible Link Functions in Nonparametric Binary Regression with Gaussian Process Priors

Modelling Extreme Rain Accumulation with an Application to the 2011 Lake Champlain Flood

Cross-Validatory Extreme Value Threshold Selection and Uncertainty with Application to Ocean Storm Severity

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