论文标题
通过相对$α$ -Entropy的信息几何形状
Generalized Bayesian Cramér-Rao Inequality via Information Geometry of Relative $α$-Entropy
论文作者
论文摘要
相对$α$ - 凝性是相对熵的rényi类似物，并且在信息理论问题中显着出现。有关此数量的最新信息几何研究使Cramér-Rao不平等的概括为基础参数概率分布的估计量的方差提供了下限。但是，在贝叶斯框架中，该框架仍未进行。在本文中，我们提出了一个基于相对$α$ entropy的一般riemannian公制，以获得广义的贝叶斯cramér-rao不平等。这为$α$ -ESCORT分布的无偏估计量的方差确定了从基础分布的无偏估计器开始。我们表明，在熵顺序接近统一的限制案例中，该框架将减少到常规的贝叶斯cramér-rao不平等。此外，在没有先验的情况下，相同的框架产生了确定性的cramér-rao不平等。
The relative $α$-entropy is the Rényi analog of relative entropy and arises prominently in information-theoretic problems. Recent information geometric investigations on this quantity have enabled the generalization of the Cramér-Rao inequality, which provides a lower bound for the variance of an estimator of an escort of the underlying parametric probability distribution. However, this framework remains unexamined in the Bayesian framework. In this paper, we propose a general Riemannian metric based on relative $α$-entropy to obtain a generalized Bayesian Cramér-Rao inequality. This establishes a lower bound for the variance of an unbiased estimator for the $α$-escort distribution starting from an unbiased estimator for the underlying distribution. We show that in the limiting case when the entropy order approaches unity, this framework reduces to the conventional Bayesian Cramér-Rao inequality. Further, in the absence of priors, the same framework yields the deterministic Cramér-Rao inequality.
