Bayesian Model Checking with Applications to Hierarchical Models

Bayesian Model Checking with Applications to Hierarchical Models Robert E. Weiss* August 13, 1996 Abstract In a Bayesian model with proper prior, all functions of the parameters and data are known. After observing the data, the joint prior specification of data and parameters can be checked by comparing the posterior of any function of the parameters to its assumed prior. This paper gives checks for missing predictors, goodness-of-fit, and over-diffuseness of the prior. The approach is illustrated in a hierarchical random effects model. Key Words: Bayesian Data Analysis, Diagnostics, Goodness-of-Fit, Longitu- dinal Data, Outlier, Quantile-Quantile Plots. Introduction. This paper introduces a general approach to Bayesian model checking. Like pre- vious authors (Box, 1981; Chaloner and Brant 1988; Dey, Gelfand, Vlachos and Schwarz 1994; Gelman, Meng and Stern 1996; Meng 1994; Rubin 1984), we may consider a model suspect when some residual or checking function g, a function of the data Y and/or parameters 6, is far from an appropriate measure of center, * Robert E. Weiss is Assistant Professor, Department of Biostatistics, Box 177220; UCLA School of Public Health; Los Angeles CA 90095-1772 U.S.A.; email rob@rem.ph.ucla.edu. This work was supported by grant #GM50011 from the NIGM. The author thanks Charlie Zhang and Meehyung Cho for help with the calculations and graphs and M. Cho and E. Bradlow for comments.