Case influence analysis in Bayesian inference

We demonstrate how case influence analysis. commonly used in regression, can be applied to Bayesian hierarchical models. Draws from the joint posterior distribution of parameters are importance weighted to reflect the effect of deleting each observation in turn: the ensuing changes in the posterior distribution of each parameter are displayed graphically. The procedure is particularly useful when drawing a sample from the posterior distribution requires extensive calculations (as with a Markov Chain Monte Carlo sampler). The structure of hierarchical models, and other models with local dependence, makes the importance weights inexpensive to calculate with little additional programming. Some new alternative weighting schemes are described that extend the range of problems in which reweighting can be used to assess influence. Applications: to a growth curve model and a complex hierarchical model for opinion data are described. Our focus on case influence on parameters is complementary to other work that measures influence by distances between posterior or predictive distributions.