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For performing regression tasks involved in various physics problems, enhancing the precision or equivalently reducing the uncertainty of regression results is undoubtedly one of the central goals. Here, somewhat surprisingly, we find that the unfavorable regression uncertainty in performing the regression tasks of inverse statistical problems actually contains hidden information concerning the phase transitions of the system under consideration. By utilizing this hidden information, we develop a new unsupervised machine learning approach for automated detection of phases of matter, dubbed learning from regression uncertainty. This is achieved by revealing an intrinsic connection between regression uncertainty and response properties of the system, thus making the outputs of this machine learning approach directly interpretable via conventional notions of physics. We demonstrate the approach by identifying the critical points of the ferromagnetic Ising model and the three-state clock model, and revealing the existence of the intermediate phase in the six-state and seven-state clock models. Comparing to the widely-used classification-based approaches developed so far, although successful, their recognized classes of patterns are essentially abstract, which hinders their straightforward relation to conventional notions of physics. These challenges persist even when one employs the state-of-the-art deep neural networks that excel at classification tasks. In contrast, with the core working horse being a neural network performing regression tasks, our new approach is not only practically more efficient, but also paves the way towards intriguing possibilities for unveiling new physics via machine learning in a physically interpretable manner.