Gauss' law of error is generalized in Tsallis statistics such as multifractal systems, in which Tsallis entropy plays an essential role instead of Shannon entropy. For the generalization, we apply the new multiplication operation determined by the q−logarithm and the q−exponential functions to the definition of the likelihood function in Gauss' law of error. The maximum likelihood principle leads us to finding Tsallis distribution as nonextensively generalization of Gaussian distribution.

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The Shannon-Khinchin axioms are generalized to nonextensive systems and the uniqueness theorem for the nonextensive entropy is proved rigorously. In the present axioms, Shannon additivity is used as additivity in contrast to pseudoadditivity in Abe's axioms. The results reveal that Tsallis entropy is the simplest among all nonextensive entropies which can be obtained from the generalized Shannon-Khinchin axioms. * Electronic address: suyari@tj.chiba-u.ac.jp

Based on the one-parameter generalization of Shannon-Khinchin (SK) axioms presented by one of the authors, and utilizing a tree-graphical representation, we have further developed the SK Axioms in accordance with the two-parameter entropy introduced by Sharma-Taneja, Mittal, Borges-Roditi, and Kaniadakis-Lissia-Scarfone. The corresponding unique theorem is proved. It is shown that the obtained two-parameter Shannon additivity is a natural consequence from the Leibniz rule of the two-parameter Chakrabarti-Jagannathan difference operator.

Study Design. Retrospective analysis of magnetic resonance imaging (MRI). Objective. The aim of this study was to evaluate the performance of our convolutional neural network (CNN) in differentiating between spinal schwannoma and meningioma on MRI. We compared the performance of the CNN and that of two expert radiologists. Summary of Background Data. Preoperative discrimination between spinal schwannomas and meningiomas is crucial because different surgical procedures are required for their treatment. A deep-learning approach based on CNNs is gaining interest in the medical imaging field. Methods. We retrospectively reviewed data from patients with spinal schwannoma and meningioma who had undergone MRI and tumor resection. There were 50 patients with schwannoma and 34 patients with meningioma. Sagittal T2-weighted magnetic resonance imaging (T2WI) and sagittal contrast-enhanced T1-weighted magnetic resonance imaging (T1WI) were used for the CNN training and validation. The deep learning framework Tensorflow was used to construct the CNN architecture. To evaluate the performance of the CNN, we plotted the receiver-operating characteristic (ROC) curve and calculated the area under the curve (AUC). We calculated and compared the sensitivity, specificity, and accuracy of the diagnosis by the CNN and two board-certified radiologists. Results. . The AUC of ROC curves of the CNN based on T2WI and contrast-enhanced T1WI were 0.876 and 0.870, respectively. The sensitivity of the CNN based on T2WI was 78%; 100% for radiologist 1; and 95% for radiologist 2. The specificity was 82%, 26%, and 42%, respectively. The accuracy was 80%, 69%, and 73%, respectively. By contrast, the sensitivity of the CNN based on contrast-enhanced T1WI was 85%; 100% for radiologist 1; and 96% for radiologist 2. The specificity was 75%, 56, and 58%, respectively. The accuracy was 81%, 82%, and 81%, respectively. Conclusion. We have successfully differentiated spinal schwannomas and meningiomas using the CNN with high diagnostic accuracy comparable to that of experienced radiologists. Level of Evidence: 4