In this work some advances in the theory of curvature of two-dimensional probability manifolds corresponding to families of distributions are proposed. It is proved that location-scale distributions are hyperbolic in the Information Geometry sense even when the generatrix is non-even or non-smooth. A novel formula is obtained for the computation of curvature in the case of exponential families: this formula implies some new flatness criteria in dimension 2. Finally, it is observed that many two parameter distributions, widely used in applications, are locally hyperbolic, which highlights the role of hyperbolic geometry in the study of commonly employed probability manifolds. These results have benefited from the use of explainable computational tools, which can substantially boost scientific productivity.

翻译：本文提出了若干关于分布族对应的二维概率流形曲率理论的进展。证明了即便母线为非对称或非光滑的情形，位置-尺度分布在信息几何意义下仍是双曲型的。针对指数族情形，本文推导出一个新的曲率计算公式，该公式蕴含了若干二维情形下新的平坦性判据。最后，观察到许多实际应用中广泛使用的双参数分布具有局部双曲性，这凸显了双曲几何在研究常见概率流形中的重要作用。这些成果得益于可解释性计算工具的应用，此类工具能够显著提升科研产出效率。