Different data types can offer complementary perspectives on the same biological phenomenon. In cancer studies, for example, data on copy number alterations indicate losses and amplifications of genomic regions in tumours, while transcriptomic data point to the impact of genomic and environmental events on the internal wiring of the cell. Fusing different data provides a more comprehensive model of the cancer cell than that offered by any single type. However, biological signals in different patients exhibit diverse degrees of concordance due to cancer heterogeneity and inherent noise in the measurements. This is a particularly important issue in cancer subtype discovery, where personalised strategies to guide therapy are of vital importance. We present a nonparametric Bayesian model for discovering prognostic cancer subtypes by integrating gene expression and copy number variation data. Our model is constructed from a hierarchy of Dirichlet Processes and addresses three key challenges in data fusion: (i) To separate concordant from discordant signals, (ii) to select informative features, (iii) to estimate the number of disease subtypes. Concordance of signals is assessed individually for each patient, giving us an additional level of insight into the underlying disease structure. We exemplify the power of our model in prostate cancer and breast cancer and show that it outperforms competing methods. In the prostate cancer data, we identify an entirely new subtype with extremely poor survival outcome and show how other analyses fail to detect it. In the breast cancer data, we find subtypes with superior prognostic value by using the concordant results. These discoveries were crucially dependent on our model's ability to distinguish concordant and discordant signals within each patient sample, and would otherwise have been missed. We therefore demonstrate the importance of taking a patient-specific approach, using highly-flexible nonparametric Bayesian methods.