The hyperbolic secant distribution has several generalizations with applications in finance. In this study, we explore the dual geometric structure of one such generalization, namely the beta-logistic distribution. Recent findings also interpret Bernoulli and Euler polynomials as moments of specific random variables, treating them as special cases within the framework of the beta-logistic distribution. The current study also uncovers that the beta-logistic distribution admits an $\alpha$-parallel prior for any real number $\alpha$, that has the potential for application in geometric statistical inference.

翻译：双曲正割分布有多个适用于金融领域的推广形式。本研究探讨了其中一种推广形式——beta-logistic分布的对偶几何结构。最新研究还将伯努利多项式与欧拉多项式解释为特定随机变量的矩，并将其作为beta-logistic分布框架下的特例处理。本研究同时发现，beta-logistic分布对任意实数$\alpha$均具备$\alpha$-平行先验，这一性质在几何统计推断中具有潜在应用价值。