Influence Measures for Bayesian Data Analysis

Identifying influential observations in the data is desired to ensure proper inference and statistical analysis. Modern methods to identify influence cases uses cross-validation diagnostics based on the effect of deletion of i-th observation on inference. A popular method to identify influential observations is to use Kullback-Liebler divergence measure between the posterior distribution of the parameter of interest given full data and the posterior distribution given the cross-validated data, where the cross-validated data has the i-th observation removed. Although, in Bayesian inference, the posterior distribution contains all the relevant information about a parameter of interest, when the goal is prediction, perhaps the predictive distribution should be used to identifying influential observations. So, we extended our method to the comparison of the posterior predictive distributions given full data and cross-validated data. We generalize and extend existing popular Bayesian cross-validated influence diagnostics using Bregman divergence based measure (BD). We derive useful properties of these BD based on the influence of each observation on the posterior distribution and we show that it can be extended to the predictive distribution. We show that these BD based measures allow interpretable calibration and that they can be computed via Monte Carlo Markov Chain (MCMC) samples from a single posterior based on full data. We illustrate how our new measure of influence of observations have more useful practical roles for data analysis than popular Bayesian residual analysis tools (CPO) in an example of meta-analysis with binary response and in other cases of interval-censored data.