α-parallel prior and its properties
α-parallel prior and its properties
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DOI：
10.1109/tit.2004.842703
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发表时间：
2005-03-01
影响因子：
2.5
通讯作者：
Amari, S
中科院分区：
文献类型：
--
作者：
Takeuchi, J;Amari, S
Jeffreys先验在统计推断中起着重要的作用。本文从信息几何的角度对Jeffreys先验进行了推广，引入了一类单参数先验分布族，我们称之为α-平行先验。α-平行先验被定义为相对于α-连接的平行体积元，并且当α = 0时与Jeffreys先验一致。此外，我们分析的渐近行为的各种估计，如投影贝叶斯估计（估计通过投影贝叶斯预测密度到原始类的分布）和最小描述长度（MDL）估计，当使用的a-平行先验。这些估计的差异，从最大似然估计（MLE）由于阿尔法先验被证明是由一个不变的向量场的统计模型。虽然Jeffreys先验总是存在的，但不总是保证存在α不等于0的α平行先验。因此，我们认为存在的a-平行先验的条件，阐明共轭对称的统计模型。
It is known that the Jeffreys prior plays an important role in statistical inference. In this paper, we generalize the Jeffreys prior from the point of view of information geometry and introduce a one-parameter family of prior distributions, which we named the alpha-parallel priors. The alpha-parallel prior is defined as the parallel volume element with respect to the alpha-connection and coincides with the Jeffreys prior when alpha = 0. Further, we analyze asymptotic behavior of the various estimators such as the projected Bayes estimator (the estimator obtained by projecting the Bayes predictive density onto the original class of distributions) and the minimum description length (MDL) estimator, when the a-parallel prior is used. The difference of these estimators from maximum-likelihood estimator (MLE) due to the alpha-prior is shown to be regulated by an invariant vector field of the statistical model. Although the Jeffreys prior always exists, the existence of alpha-parallel prior with a not equal 0 is not always guaranteed. Hence, we consider conditions for the existence of the a-parallel prior, elucidating the conjugate symmetry in a statistical model.