相对Fisher信息的Legendre变换结构及其极值性质
Legendre transform structure and extremal properties of the relative Fisher information
摘要：
进行相对Fisher信息（RFI，下文）的变分极值化。互易关系，类似于热力学的推导，采用极值结果的RFI表示的概率幅。推导了RFI的时间无关的类薛定谔方程（类薛定谔链路）。伴随的勒让德变换结构（LTS，下文）是通过利用广义RFI-Euler定理，这表明，整个热力学的数学结构转化为RFI框架，无论是平衡和非平衡的情况下开发。定性不同性质的本结果\textit{vis-á-vis}那些以前的研究利用香农熵和/或Fisher信息测度（ESTA，下文）进行了讨论。一个原则之间的关系RFI和可持续发展框架的推导。这种关系的效用证明了一个例子，其中的能量本征值的薛定谔链路的RFI是推断单独使用量子力学维里定理和LTS的RFI。
英文摘要：
Variational extremization of the relative Fisher information (RFI, hereafter) is performed. Reciprocity relations, akin to those of thermodynamics are derived, employing the extremal results of the RFI expressed in terms of probability amplitudes. A time independent Schrödinger-like equation (Schrödinger-like link) for the RFI is derived. The concomitant Legendre transform structure (LTS, hereafter) is developed by utilizing a generalized RFI-Euler theorem, which shows that the entire mathematical structure of thermodynamics translates into the RFI framework, both for equilibrium and non-equilibrium cases. The qualitatively distinct nature of the present results \textit{vis-á-vis} those of prior studies utilizing the Shannon entropy and/or the Fisher information measure (FIM, hereafter) is discussed. A principled relationship between the RFI and the FIM frameworks is derived. The utility of this relationship is demonstrated by an example wherein the energy eigenvalues of the Schrödinger-like link for the RFI is inferred solely using the quantum mechanical virial theorem and the LTS of the RFI.
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