Multiple imputation is widely used for handling missing data in real-world applications. For variable selection on multiply-imputed datasets, however, if selection is performed on each imputed dataset separately, it can result in different sets of selected variables across datasets. MI-LASSO, one of the most commonly used approaches to this problem, regards the same variable across all separate imputed datasets as a group variable and exploits the group LASSO to yield a consistent variable selection across all the multiply-imputed datasets. In this paper, we extend MI-LASSO into the Bayesian framework and propose four Bayesian MI-LASSO models for variable selection on multiply-imputed data, including three shrinkage prior-based and one Spike-and-Slab prior-based methods. To overcome the limitations of traditional, ad hoc threshold-based model selection, we develop a principled four-step projection predictive procedure that provides robust variable selection. This procedure also facilitates approximately calibrated post-selection inference and attains near-nominal coverage under a correct selection model. Simulation studies showed that the Bayesian MI-LASSO outperformed MI-LASSO and other alternative approaches, achieving higher specificity and lower mean squared error across a range of settings. We further demonstrated these methods via a case study using a multiply-imputed dataset from the University of Michigan Dioxin Exposure Study. The R package BMIselect is available on CRAN.

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