Predictive algorithms inform consequential decisions in settings where the outcome is selectively observed given some choices made by human decision makers. There often exists unobserved confounders that affected the decision maker's choice and the outcome. We propose a unified methodology for the robust design and evaluation of predictive algorithms in selectively observed data under such unobserved confounding. Our approach imposes general assumptions on how much the outcome may vary on average between unselected and selected units conditional on observed covariates and identified nuisance parameters, formalizing popular empirical strategies for imputing missing data such as proxy outcomes and instrumental variables. We develop debiased machine learning estimators for the bounds on a large class of predictive performance estimands, such as the conditional likelihood of the outcome, a predictive algorithm's mean square error, true/false positive rate, and many others, under these assumptions. In an administrative dataset from a large Australian financial institution, we illustrate how varying assumptions on unobserved confounding leads to meaningful changes in default risk predictions and evaluations of credit scores across sensitive groups.

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