Reparameterization of extreme value framework for improved Bayesian workflow. (English) Zbl 07737941
Summary: Using Bayesian methods for extreme value analysis offers an alternative to frequentist ones, with several advantages such as easily dealing with parametric uncertainty or studying irregular models. However, computations can be challenging and the efficiency of algorithms can be altered by poor parametrization choices. The focus is on the Poisson process characterization of univariate extremes and outline two key benefits of an orthogonal parameterization. First, Markov chain Monte Carlo convergence is improved when applied on orthogonal parameters. This analysis relies on convergence diagnostics computed on several simulations. Second, orthogonalization also helps deriving Jeffreys and penalized complexity priors, and establishing posterior propriety thereof. The proposed framework is applied to return level estimation of Garonne flow data (France).
MSC:
| 62-08 | Computational methods for problems pertaining to statistics |
References:
| [1] | Albert, C., Estimation des limites d’extrapolation par les lois de valeurs extrêmes. Application à des données environnementales (2018), Université Grenoble Alpes, PhD thesis (in French) |
| [2] | Albert, C.; Dutfoy, A.; Gardes, L.; Girard, S., An extreme quantile estimator for the log-generalized Weibull-tail model, Econom. Stat., 13, 137-174 (2020) |
| [3] | Belzile, L. R.; Dutang, C.; Northrop, P. J.; Opitz, T., A modeler’s guide to extreme value software (2022) |
| [4] | Betancourt, M., Incomplete reparameterizations and equivalent metrics (2019) |
| [5] | Betancourt, M.; Girolami, M., Hamiltonian Monte Carlo for hierarchical models, (Current Trends in Bayesian Methodology with Applications (2015), CRC Press), 79-101 |
| [6] | Bousquet, N., Bayesian extreme value theory, (Extreme Value Theory with Applications to Natural Hazards: From Statistical Theory to Industrial Practice (2021), Springer: Springer Cham), 271-325 |
| [7] | Browne, W. J.; Steele, F.; Golalizadeh, M.; Green, M. J., The use of simple reparameterizations to improve the efficiency of Markov chain Monte Carlo estimation for multilevel models with applications to discrete time survival models, J. R. Stat. Soc. A, 172, 579-598 (2009) |
| [8] | Castellanos, M. E.; Cabras, S., A default Bayesian procedure for the generalized Pareto distribution, J. Stat. Plan. Inference, 137, 473-483 (2007) · Zbl 1102.62023 |
| [9] | Chavez-Demoulin, V.; Davison, A. C., Generalized additive modelling of sample extremes, J. R. Stat. Soc., Ser. C, 54, 207-222 (2005) · Zbl 1490.62194 |
| [10] | Coles, S., An Introduction to Statistical Modeling of Extreme Values, Springer Series in Statistics (2001), Springer-Verlag: Springer-Verlag London · Zbl 0980.62043 |
| [11] | Coles, S. G.; Powell, E. A., Bayesian methods in extreme value modelling: a review and new developments, Int. Stat. Rev., 64, 119-136 (1996) · Zbl 0853.62025 |
| [12] | Cox, D. R.; Reid, N., Parameter orthogonality and approximate conditional inference, J. R. Stat. Soc., Ser. B, 49, 1-18 (1987) · Zbl 0616.62006 |
| [13] | Diebolt, J.; El-Aroui, M. A.; Garrido, M.; Girard, S., Quasi-conjugate Bayes estimates for GPD parameters and application to heavy tails modelling, Extremes, 8, 57-78 (2005) · Zbl 1091.62009 |
| [14] | Fawcett, L.; Green, A. C., Bayesian posterior predictive return levels for environmental extremes, Stoch. Environ. Res. Risk Assess., 32, 2233-2252 (2018) |
| [15] | Gelfand, A. E.; Sahu, S. K.; Carlin, B. P., Efficient parametrisations for normal linear mixed models, Biometrika, 82, 479-488 (1995) · Zbl 0832.62064 |
| [16] | Gelfand, A. E.; Sahu, S. K.; Carlin, B. P., Efficient parametrizations for generalized linear mixed models, Bayesian Stat., 5, 48-74 (1996) |
| [17] | Gelman, A.; Carlin, J. B.; Stern, H. S.; Dunson, D. B.; Vehtari, A.; Rubin, D. B., Bayesian Data Analysis (2013), CRC Press: CRC Press New York |
| [18] | Gelman, A.; Vehtari, A.; Simpson, D.; Margossian, C. C.; Carpenter, B.; Yao, Y.; Kennedy, L.; Gabry, J.; Bürkner, P. C.; Modrák, M., Bayesian workflow (2020) |
| [19] | Gilks, W. R.; Richardson, S.; Spiegelhalter, D., Markov Chain Monte Carlo in Practice (1995), CRC Press: CRC Press Boca Raton · Zbl 0832.00018 |
| [20] | Gilleland, E.; Katz, R. W., extRemes 2.0: an extreme value analysis package in R, J. Stat. Softw., 72, 1-39 (2016) |
| [21] | Haan, L.; Ferreira, A., Extreme Value Theory: an Introduction (2006), Springer: Springer New York · Zbl 1101.62002 |
| [22] | Hoffman, M. D.; Gelman, A., No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo, J. Mach. Learn. Res., 15, 1593-1623 (2014) · Zbl 1319.60150 |
| [23] | Huzurbazar, V. S., Probability distributions and orthogonal parameters, Math. Proc. Camb. Philos. Soc., 46, 281-284 (1950) · Zbl 0036.09304 |
| [24] | Jeffreys, H., An invariant form for the prior probability in estimation problems, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., 186, 453-461 (1946) · Zbl 0063.03050 |
| [25] | Jeffreys, H., The Theory of Probability (1961), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0116.34904 |
| [26] | Jóhannesson, Á. V.; Siegert, S.; Huser, R.; Bakka, H.; Hrafnkelsson, B., Approximate Bayesian inference for analysis of spatiotemporal flood frequency data, Ann. Appl. Stat., 16, 905-935 (2022) · Zbl 1498.62285 |
| [27] | Jonathan, P.; Randell, D.; Wadsworth, J.; Tawn, J., Uncertainties in return values from extreme value analysis of peaks over threshold using the generalised Pareto distribution, Ocean Eng., 220, Article 107725 pp. (2021) |
| [28] | Kotz, S.; Nadarajah, S., Extreme Value Distributions: Theory and Applications (2000), Imperial College Press: Imperial College Press London · Zbl 0960.62051 |
| [29] | Leadbetter, M.; Lindgren, G.; Rootzén, H., Extremes and Related Properties of Random Sequences and Processes (1983), Springer: Springer New York · Zbl 0518.60021 |
| [30] | Moins, T.; Arbel, J.; Dutfoy, A.; Girard, S., On the use of a local \(\hat{R}\) to improve MCMC convergence diagnostic (2023) |
| [31] | Neal, R. M., MCMC using Hamiltonian dynamics, (Handbook of Markov Chain Monte Carlo (2011), Chapman and Hall/CRC), 113-162 · Zbl 1229.65018 |
| [32] | Northrop, P. J., rust: Ratio-of-Uniforms Simulation with Transformation (2022) |
| [33] | Northrop, P. J., revdbayes: Ratio-of-Uniforms Sampling for Bayesian Extreme Value Analysis (2023) |
| [34] | Northrop, P. J.; Attalides, N., Posterior propriety in Bayesian extreme value analyses using reference priors, Stat. Sin., 26, 721-743 (2016) · Zbl 1360.62247 |
| [35] | Opitz, T.; Huser, R.; Bakka, H.; Rue, H., INLA goes extreme: Bayesian tail regression for the estimation of high spatio-temporal quantiles, Extremes, 21, 441-462 (2018) · Zbl 1407.62167 |
| [36] | Papaspiliopoulos, O.; Roberts, G. O.; Sköld, M., Non-centered parameterisations for hierarchical models and data augmentation, Bayesian Stat., 7, 307-326 (2003) |
| [37] | Pickands, J., Statistical inference using extreme order statistics, Ann. Stat., 3, 119-131 (1975) · Zbl 0312.62038 |
| [38] | Robert, C. P., The Bayesian Choice: from Decision-Theoretic Foundations to Computational Implementation (2007), Springer: Springer New York · Zbl 1129.62003 |
| [39] | Roberts, G. O.; Polson, N. G., On the geometric convergence of the Gibbs sampler, J. R. Stat. Soc., Ser. B, 56, 377-384 (1994) · Zbl 0796.62029 |
| [40] | Roberts, G. O.; Sahu, S. K., Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler, J. R. Stat. Soc., Ser. B, 59, 291-317 (1997) · Zbl 0886.62083 |
| [41] | Salvatier, J.; Wiecki, T. V.; Fonnesbeck, C., Probabilistic programming in Python using PyMC3, PeerJ Comput. Sci., 2, e55 (2016) |
| [42] | Sharkey, P.; Tawn, J. A., A Poisson process reparameterisation for Bayesian inference for extremes, Extremes, 20, 239-263 (2017) · Zbl 1373.60091 |
| [43] | Simpson, D.; Rue, H.; Riebler, A.; Martins, T. G.; Sørbye, S. H., Penalising model component complexity: a principled, practical approach to constructing priors, Stat. Sci., 32, 1-28 (2017) · Zbl 1442.62060 |
| [44] | Stephenson, A., Bayesian inference for extreme value modelling, (Extreme Value Modeling and Risk Analysis: Methods and Applications (2016), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, Florida), 257-280 · Zbl 1365.62100 |
| [45] | Stephenson, A.; Tawn, J., Bayesian inference for extremes: accounting for the three extremal types, Extremes, 7, 291-307 (2004) · Zbl 1090.62025 |
| [46] | Tibshirani, R.; Wasserman, L., Some aspects of the reparametrization of statistical models, Can. J. Stat., 22, 163-173 (1994) · Zbl 0802.62036 |
| [47] | Van der Vaart, A. W., Asymptotic Statistics (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0910.62001 |
| [48] | Wadsworth, J. L.; Tawn, J. A.; Jonathan, P., Accounting for choice of measurement scale in extreme value modeling, Ann. Appl. Stat., 4, 1558-1578 (2010) · Zbl 1202.62066 |
| [49] | Woutersen, T., Consistent estimation and orthogonality, Adv. Econom., 27, 155-178 (2011) · Zbl 1443.62037 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
