基于平衡图的Infinite-dimensional统计流形
Infinite-dimensional statistical manifolds based on a balanced chart
摘要：
我们开发了一个家庭的无限维Banach流形上的抽象可测空间的措施，采用图表之间的密度和日志密度函数的“平衡”。流形$(\tilde{M}_λ,λ\in [2,\infty))$保留了有限维信息几何的许多特征;特别是$α$ -发散属于$C^{\lceilλ\rceil-1}$类，使得能够定义Fisher度量和$α$ -特定类向量场的导数。基于图表中心版本的概率测度流形$(M_λ,λ\in [2,\infty))$被证明是$C^{\lceilλ\rceil-1}$--$\tilde{M}_λ$的嵌入子流形。Fisher度量是$\tilde{M}_λ$上的伪黎曼度量。然而，当限制到有限维嵌入子流形，它成为黎曼度量，允许充分发展的几何$α$ -共变导数。$\tilde{M}_λ$和$M_λ$为贝叶斯估计问题中后验分布的近似值的研究和比较提供了自然设置。
英文摘要：
We develop a family of infinite-dimensional Banach manifolds of measures on an abstract measurable space, employing charts that are "balanced" between the density and log-density functions. The manifolds, $(\tilde{M}_λ,λ\in [2,\infty))$, retain many of the features of finite-dimensional information geometry; in particular, the $α$-divergences are of class $C^{\lceilλ\rceil-1}$, enabling the definition of the Fisher metric and $α$-derivatives of particular classes of vector fields. Manifolds of probability measures, $(M_λ,λ\in [2,\infty))$, based on centred versions of the charts are shown to be $C^{\lceilλ\rceil-1}$-embedded submanifolds of the $\tilde{M}_λ$. The Fisher metric is a pseudo-Riemannian metric on $\tilde{M}_λ$. However, when restricted to finite-dimensional embedded submanifolds it becomes a Riemannian metric, allowing the full development of the geometry of $α$-covariant derivatives. $\tilde{M}_λ$ and $M_λ$ provide natural settings for the study and comparison of approximations to posterior distributions in problems of Bayesian estimation.
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