Information geometry is concerned with the application of differential geometry concepts in the study of the parametric spaces of statistical models. When the random variables are independent and identically distributed, the underlying parametric space exhibit constant curvature, which makes the geometry hyperbolic (negative) or spherical (positive). In this paper, we derive closed-form expressions for the components of the first and second fundamental forms regarding pairwise isotropic Gaussian-Markov random field manifolds, allowing the computation of the Gaussian, mean and principal curvatures. Computational simulations using Markov Chain Monte Carlo dynamics indicate that a change in the sign of the Gaussian curvature is related to the emergence of phase transitions in the field. Moreover, the curvatures are highly asymmetrical for positive and negative displacements in the inverse temperature parameter, suggesting the existence of irreversible geometric properties in the parametric space along the dynamics. Furthermore, these asymmetric changes in the curvature of the space induces an intrinsic notion of time in the evolution of the random field.

翻译：信息几何与统计模型参数空间研究中差异几何概念的应用有关。当随机变量是独立且分布相同的时,基本参数空间呈现出恒定的曲度,这使得地球物理超偏(负)或球形(正)形成。在本文中,我们为第一和第二种基本形式中关于对等等等异质高斯-马尔科夫随机场数的构件得出了封闭式表达式,从而可以计算高斯、平均和主要曲度。使用Markov链蒙特卡洛动态的计算模拟表明,高斯曲线的标志变化与实地阶段转变的出现有关。此外,曲线对于反向温度参数的正偏移和负移而言,高度不对称,表明在与动态相近的对等空间存在不可逆转的几何几何特征。此外,空间曲度的这些不对称变化还引出了随机场演变时间的内在概念。