One of the dramatic trends at the intersection of computing and healthcare has been patients' increased access to medical information, ranging from self-tracked physiological data to genetic data, tests, and scans. Increasingly however, patients and clinicians have access to advanced machine learning-based tools for diagnosis, prediction, and recommendation based on large amounts of data, some of it patient-generated. Consequently, just as organizations have had to deal with a "Bring Your Own Device" (BYOD) reality in which employees use their personal devices (phones and tablets) for some aspects of their work, a similar reality of "Bring Your Own Algorithm" (BYOA) is emerging in healthcare with its own challenges and support demands. BYOA is changing patient-clinician interactions and the technologies, skills and workflows related to them. In this paper we argue that: (1) BYOA is changing the patient-clinician relationship and the nature of expert work in healthcare, and (2) better patient-clinician-information-interpretation relationships can be facilitated with solutions that integrate technological and organizational perspectives.