Model Selection when multiple imputation is used to protect confidentiality in public use data

Sev eral statistical agencies use, or are considering the use of, multi- ple imputation to limit the risk of disclosing respondents' identities or sensitive attributes in public use data les. For example, agencies can release partially syn- thetic datasets, comprising the units originally surveyed with some values, such as sensitive values at high risk of disclosure or values of key identiers, replaced with multiple imputations. We describe how secondary analysts of such multiply- imputed datasets can implement Bayesian model selection procedures that appro- priately condition on the multiple datasets and the information released by the agency about the imputation models. We illustrate by deriving Bayes factor ap- proximations and a data augmentation step for stochastic search variable selection algorithms.