A key challenge in medicine is accurate and timely medical diagnosis. Too often, patients do not receive the right diagnosis and as a result, fail to receive the best treatments. As an example, the Acute Respiratory Distress Syndrome (ARDS) is a severe acute lung condition that leads to high mortality. Yet, patients with ARDS frequently go unrecognized and are not given treatments that can improve their outcomes. Machine learning techniques employed on detailed electronic health data may be utilized for the prediction and early diagnosis of ARDS, helping clinicians better care for these patients. Diagnosis of ARDS may serve as a model for many medical conditions where simple, inexpensive gold standard tests are not routinely available and diagnostic uncertainty is common, even among experts. In addressing the challenge of the diagnosis of ARDS, we will explore two emerging fields of machine learning, learning from uncertain data and learning with privileged information, both of potentially high relevance to healthcare applications. While the current work focuses on diagnosis of ARDS, these learning approaches are likely to generalize across healthcare settings.