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We develop new quasi-experimental tools to understand algorithmic discrimination and build non-discriminatory algorithms when the outcome of interest is only selectively observed. These tools are applied in the context of pretrial bail decisions, where conventional algorithmic predictions are generated using only the misconduct outcomes of released defendants. We first show that algorithmic discrimination arises in such settings when the available algorithmic inputs are systematically different for white and Black defendants with the same objective misconduct potential. We then show how algorithmic discrimination can be eliminated by measuring and purging these conditional input disparities. Leveraging the quasi-random assignment of bail judges in New York City, we find that our new algorithms not only eliminate algorithmic discrimination but also generate more accurate predictions by correcting for the selective observability of misconduct outcomes.