In this paper, we investigate a duality between Hermitian and almost Kähler structures on the tangent manifold $T\mathcal{M}$ induced by pairs of conjugate connections on its base, affine Riemannian manifold $\mathcal{M}$. In the context of information geometry, the classical theory of statistical manifold (which we call $\mathcal{S}$-geometry) prescribes a parametrized family of probability distributions with a Fisher-Rao metric g and, using the Amari-Chensov tensor, a family of dualistic, torsion-free connections ${\mathrm{\nabla }}^{(\alpha )}$, known as α-connections on $\mathcal{M}$. Here we prescribe an alternative geometric framework (which we call $\mathcal{P}$-geometry or partially flat geometry) by treating such parametrization as affine coordinates with respect to a flat connection ∇, and considering its g-conjugate connection ${\mathrm{\nabla }}^{⁎}$ which is curvature-free but generally carries torsion. Under $\mathcal{P}$-geometry, the triplet $(g,\mathrm{\nabla },{\mathrm{\nabla }}^{⁎})$ on $\mathcal{M}$ leads to a pair of complex and almost Kähler structures on $T\mathcal{M}$, in “mirror correspondence” to each other. Such complex-to-symplectic correspondence is reminiscent of mirror symmetry in string theory. We discuss the statistical meaning of mirror correspondence in terms of reference duality and representation duality in (various generalizations of) contrast/divergence functions characterizing proximity of probability distributions within a parametric statistical model.

在本文中，我们研究切线流形上的埃尔米特结构与几乎Kähler结构之间的对偶性 $Ť\mathcal{中号}$ 由其仿射黎曼流形上的共轭连接对诱导 $\mathcal{中号}$。在信息几何学中，经典的统计流形理论（我们称之为$\mathcal{小号}$（-geometry）规定参数化的概率分布族，具有Fisher-Rao度量g，并使用Amari-Chensov张量，实现二元，无扭转连接${\mathrm{\nabla }}^{（\alpha ）}$，称为α连接$\mathcal{中号}$。在这里，我们规定了替代的几何框架（我们称之为$\mathcal{P}$-几何形状或部分平坦的几何形状），将此类参数化视为相对于平坦连接的仿射坐标，并考虑其g-共轭连接${\mathrm{\nabla }}^{⁎}$它没有曲率，但通常带有扭转力。下$\mathcal{P}$-几何，三元组 $（G，\mathrm{\nabla }，{\mathrm{\nabla }}^{⁎}）$ 上 $\mathcal{中号}$ 导致一对复杂且几乎是Kähler的结构 $Ť\mathcal{中号}$，彼此“镜像对应”。这种复杂到符号的对应关系让人联想到弦论中的镜像对称性。我们讨论对比/散度函数（的各种概括）中的参考对偶性和表示对偶性方面的镜像对应关系的统计意义，表征参数分布统计模型内概率分布的接近性。