We consider inference on parameters and hypotheses selected using p-values. We provide a simple procedure to decompose a p-value into two components, one for inference and one for selection, that are independent under the null hypothesis. To control the Type 1 error rate conditional on selection of the null hypothesis from randomized p-values, we develop tests motivated by the selective inference approaches of data thinning, fission, and carving. Unlike the typical application of these approaches, however, our tests only make use of the p-values rather than the full data. Furthermore, we construct confidence intervals that attain the nominal conditional coverage and corresponding estimators. In the context of the file drawer problem, the winner's curse, and sequential testing, we show that our proposals offer both theoretical and empirical advantages over existing randomized and non-randomized approaches. Lastly, we apply our methods for selective inference on genetic associations using publicly-available p-values from the UK Biobank.

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