Local predictive influence in bayesian linear models with conjugate priors

Cook (1986) presented the idea of local influence to study the sensitivity of inferences to model assumptions:introduce a vector δ of perturbations to the model; choose a discrepancy function D to measure differences between the original inference and the inference under the perturbed model; study the behavior of D near δ = 0, the original model, usually by taking derivatives. Johnson and Geisser (1983) measure influence in Bayesian inference by the Kullback-Leibler divergence between predictive distributions. I~IcCulloch (1989) is a synthesis of Cook and Johnson and Geisser, using Kullback-Leibler divergence between posterior or predictive distributions as the discrepancy function in Bayesian local influence analyses. We analyze a special case for which McCulloch gives the general theory; namely, the linear model with conjugate prior. We present specific formulae for local influence measures for 1) changes in the parameters of the gamma prior for the precision, 2) changes in the mean of the normal prior for the regression coefficients, 3) changes in the covariance matrix of the normal prior for the regression coefficients and 4) changes in the case weights. Our method is an easy way to find locally influential subsets of points without knowing in advance the sizes of the subsets. The techniques are illustrated with a regression example.