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Automated screening systems are commonly used to detect some agent in a sample and take a global decision about the subject (e.g., ill/healthy) based on these detections. We propose a Bayesian methodology for taking decisions in (sequential) screening systems that considers the false alarm rate of the detector. Our approach assesses the quality of its decisions and provides lower bounds on the achievable performance of the screening system from the training data. In addition, we develop a complete screening system for sputum smears in tuberculosis diagnosis, and show, using a real-world database, the advantages of the proposed framework when compared to the commonly used count detections and threshold approach.