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Posterior Predictive p ‐values in Bayesian Hierarchical Models
Author(s) -
STEINBAKK GUNNHILDUR HÖGNADÓTTIR,
STORVIK GEIR OLVE
Publication year - 2009
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2008.00630.x
Subject(s) - mathematics , bayesian probability , multinomial distribution , bayesian hierarchical modeling , posterior predictive distribution , posterior probability , calibration , range (aeronautics) , interpretation (philosophy) , limit (mathematics) , bayesian linear regression , statistics , bayesian inference , econometrics , computer science , materials science , composite material , mathematical analysis , programming language
. The present work focuses on extensions of the posterior predictive p ‐value (ppp‐value) for models with hierarchical structure, designed for testing assumptions made on underlying processes. The ppp‐values are popular as tools for model criticism, yet their lack of a common interpretation limit their practical use. We discuss different extensions of ppp‐values to hierarchical models, allowing for discrepancy measures that can be used for checking properties of the model at all stages. Through analytical derivations and simulation studies on simple models, we show that similar to the standard ppp‐values, these extensions are typically far from uniformly distributed under the model assumptions and can give poor power in a hypothesis testing framework. We propose a calibration of the p ‐values, making the resulting calibrated p ‐values uniformly distributed under the model conditions. Illustrations are made through a real example of multinomial regression to age distributions of fish.