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Computer-Aided Diagnosis (CAD) systems for melanoma detection have received a lot of attention during the last decades because of the utmost importance of detecting this type of skin cancer in its early stages. However, despite of the many research efforts devoted to this matter, these systems are not used yet in everyday clinical practice. Very likely, this is due to two main reasons: 1) the accuracy of the systems is not high enough; and 2) they simply provide a parallel diagnosis that actually does not help to the doctors (as long as there is no way to interpret it). In this paper, we propose a novel approach that aims to provide the doctor with an enriched diagnosis. Specifically, we rely on a dermoscopic-structure-based soft segmentation to design a set of structure-specific classifiers. Each individual structure-specific classifier is trained to distinguish benign lesions from melanomas just paying attention to one type of dermoscopic structure. Then, the outputs of the individual classifiers are combined by a means of the Bayesian method that, besides the final diagnosis, provide the doctor with additional valuable information, such as the opinions of the individual structure-specific experts and the uncertainty of the diagnosis. The results in terms of the features selected for the structure-specific classifiers are consistent with the expert insights. Furthermore, regarding the automatic melanoma diagnosis problem, the proposed method has been assessed on two different datasets, and the experimental results revealed that the proposed system clearly outperforms other methods in two datasets and compares well with the official submissions of the ISBI 2016 challenge on melanoma detection. Moreover, the system performance is equivalent to that of a well-known dermoscopy expert and its combination with the human diagnosis surpasses the human performance.