Before revision:  Our variational inferencemethod is shown to be uniformly valid and quantifies the uncertainty coming from both parameter estimation and data splitting. In essence , we take medians of p-values and medians of confidence intervals, resulting from many different data splits, and then adjust their nominal level to guarantee uniform validity.
After revision:  For inference , we take medians of p-values and medians of confidence intervals, resulting from many different data splits, and then adjust their nominal level to guarantee uniform validity.