Bayesian Methods for Information and Data Compatibility

We propose two new classes of Bayesian measure to investigate conflict among data sets from multiple studies. The first (\concentration ratio") is used to quantify the amount of information provided by a single data set through the comparison of the prior and its posterior distribution, or two data sets according to their corresponding posterior distributions. The second class (\dissonance") quantifies the extent of contradiction between two data sets. Both classes are based on volumes of highest density regions. They are well calibrated, supported by simulation, and computational algorithms are provided for their calculation. We illustrate these two classes in three real data applications: a benchmark dose toxicology study, a missing data study related to health effects of pollution, and a pediatric cancer study leveraging adult data.

Highest density regions have been criticized for being not invariant under reparametrization, so we explore an invariant version of highest density region and apply both classes of Bayesian measure based on it. A computational algorithm is provided and we illustrate the usage by revisiting the benchmark dose toxicology study and the pediatric cancer study.

We then propose an approach to quantifying the amount of information brought by a covariate in the regression model using our measures. The concentration ratio measure is used in the linear regression model and the dissonance measure is used in the logistic regression model.