Statistical inference more often than not involves models which are non-linear in the parameters thus leading to non-Gaussian posteriors. Many computational and analytical tools exist that can deal with non-Gaussian distributions, and empirical Gaussianisation transforms can reduce the amount of non-Gaussianity in a distribution. Alternatively, in this work, we employ methods from information geometry. The latter formulates a set of probability distributions for some given model as a manifold employing a Riemannian structure, equipped with a metric, the Fisher information. In this framework we study the differential geometrical meaning of non-Gaussianities in a higher order Fisher approximation, and their respective transformation behaviour under re-parameterisation, which corresponds to a chart transition on the statistical manifold. While weak non-Gaussianities vanish in normal coordinates in a first order approximation, one can in general not find transformations that discard non-Gaussianities globally. As an application we consider the likelihood of the supernovae distance-redshift relation in cosmology for the parameter pair ($\Omega_{\mathrm{m_0}}$, $w$). We demonstrate the connection between confidence intervals and geodesic length and demonstrate how the Lie-derivative along the degeneracy directions gives hints at possible isometries of the Fisher metric.

中文翻译：

---

宇宙推理问题中的信息几何

统计推断通常涉及参数非线性的模型，从而导致非高斯后验。许多计算和分析工具可以处理非高斯分布，经验高斯化变换可以减少分布中非高斯性的数量。或者，在这项工作中，我们采用信息几何的方法。后者将某些给定模型的一组概率分布公式化为采用黎曼结构的流形，配备了度量标准，即 Fisher 信息。在这个框架中，我们研究了高阶 Fisher 近似中非高斯的微分几何意义，以及它们在重新参数化下的各自变换行为，这对应于统计流形上的图表转换。虽然弱非高斯性在一阶近似中在法线坐标中消失，但通常无法找到全局丢弃非高斯性的变换。作为一个应用，我们考虑参数对 ($\Omega_{\mathrm{m_0}}$, $w$) 在宇宙学中超新星距离-红移关系的可能性。我们展示了置信区间和测地线长度之间的联系，并展示了沿简并方向的 Lie 导数如何暗示 Fisher 度量的可能等距。