Fisher度量作为信息度量的唯一性
The uniqueness of the Fisher metric as information metric
摘要：
本文在可分度量空间$Ω$上的有限非负测度空间$μ$的纤维空间$\mathcal{M}(Ω)$上定义了一个混合拓扑$\cup_μ\oplus^n L^n(μ)$，该空间具有Borel $σ$ -代数.定义了$\mathcal{M}(Ω)$上协变$n$ -张量场的强连续性概念.在信息度量强连续的假设下，证明了Fisher度量作为与$Ω$相关的统计模型的信息度量的唯一性.我们的证明实现了一个建议，由于Amari和Nagaoka推导出的唯一性的Fisher度量的特殊情况下证明的Chentsov通过使用一种特殊的限制程序。所得结果推广了有限样本空间统计模型上Fisher度量的单调性刻画，并补充了Ay-Jost-Lê-Schwachhöfer关于Fisher度量在充分统计量下不变性的唯一性定理.
英文摘要：
We define a mixed topology on the fiber space $\cup_μ\oplus^n L^n(μ)$ over the space $\mathcal{M}(Ω)$ of all finite non-negative measures $μ$ on a separable metric space $Ω$ provided with Borel $σ$-algebra. We define a notion of strong continuity of a covariant $n$-tensor field on $\mathcal{M}(Ω)$. Under the assumption of strong continuity of an information metric we prove the uniqueness of the Fisher metric as information metric on statistical models associated with $Ω$. Our proof realizes a suggestion due to Amari and Nagaoka to derive the uniqueness of the Fisher metric from the special case proved by Chentsov by using a special kind of limiting procedure. The obtained result extends the monotonicity characterization of the Fisher metric on statistical models associated with finite sample spaces and complement the uniqueness theorem by Ay-Jost-Lê-Schwachhöfer that characterizes the Fisher metric by its invariance under sufficient statistics.
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