On a variety of complex decision-making tasks, from doctors prescribing treatment to judges setting bail, machine learning algorithms have been shown to outperform expert human judgments. One complication, however, is that it is often difficult to anticipate the effects of algorithmic policies prior to deployment, as one generally cannot use historical data to directly observe what would have happened had the actions recommended by the algorithm been taken. A common strategy is to model potential outcomes for alternative decisions assuming that there are no unmeasured confounders (i.e., to assume ignorability). But if this ignorability assumption is violated, the predicted and actual effects of an algorithmic policy can diverge sharply. In this paper we present a flexible Bayesian approach to gauge the sensitivity of predicted policy outcomes to unmeasured confounders. In particular, and in contrast to past work, our modeling framework easily enables confounders to vary with the observed covariates. We demonstrate the efficacy of our method on a large dataset of judicial actions, in which one must decide whether defendants awaiting trial should be required to pay bail or can be released without payment.