用推广的Eaton马尔可夫链评估违约先验
Evaluating Default Priors with a Generalization of Eaton's Markov Chain
摘要：
我们考虑评估不适当的先验在一个正式的贝叶斯设置根据其使用的后果。设$Φ$是参数空间上的一类函数，考虑二次损失下$Φ$的元素估计.如果$Φ$中每个函数的形式Bayes估计是可容许的，则先验关于$Φ$是强可容许的.伊顿建立强容许性的方法是基于研究与推理设置相关的特定马尔可夫链的稳定性。在以前的工作中，根据$ϕ\in Φ$是有界的还是无界的，对此进行了不同的处理。我们介绍和研究一个新的马尔可夫链，使我们能够统一和推广现有的方法，同时扩大其潜在的适用范围。为了说明该方法，我们建立强容许条件时，模型是一个$p$维多元正态分布与未知的均值向量$θ$和先验的形式是$ν(\|θ\|^{2})dθ$。
英文摘要：
We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $Φ$ be a class of functions on the parameter space and consider estimating elements of $Φ$ under quadratic loss. If the formal Bayes estimator of every function in $Φ$ is admissible, then the prior is strongly admissible with respect to $Φ$. Eaton's method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with the inferential setting. In previous work, this was handled differently depending upon whether $ϕ\in Φ$ was bounded or unbounded. We introduce and study a new Markov chain which allows us to unify and generalize existing approaches while simultaneously broadening the scope of their potential applicability. To illustrate the method, we establish strong admissibility conditions when the model is a $p$-dimensional multivariate normal distribution with unknown mean vector $θ$ and the prior is of the form $ν(\|θ\|^{2})dθ$.
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