Reconciling Curvature and Importance Sampling Based Procedures for Summarizing Case Influence in Bayesian Models

Methods for summarizing case influence in Bayesian models take essentially two forms: (1) use common divergence measures for calculating distances between the full-data posterior and the case-deleted posterior, and (2) measure the impact of infinitesimal perturbations to the likelihood to study local case influence. Methods based on approach (1) lead naturally to considering the behavior of case-deletion importance sampling weights (the weights used to approximate samples from the case-deleted posterior using samples from the full posterior). Methods based on approach (2) lead naturally to considering the local curvature of the Kullback–Leibler divergence of the full posterior from a geometrically perturbed quasi-posterior. By examining the connections between the two approaches, we establish a rationale for employing low-dimensional summaries of case influence obtained entirely via the variance–covariance matrix of the log importance sampling weights. We illustrate the use of the proposed diagnostics using real and simulated data. Supplementary materials are available online.