A representation of the posterior mean for a location model
A representation of the posterior mean for a location model
复制标题
位置模型后验均值的表示
DOI：
10.1093/biomet/78.2.426
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发表时间：
1991
期刊：
影响因子：
2.7
通讯作者：
Nicholas G. Polson
中科院分区：
文献类型：
--
作者：
Nicholas G. Polson
总结后验均值的精确表示的位置模型和类的先验是正常的规模的混合物。结果利用了极大似然估计的条件分布和Masreliez定理。指出了今后的发展方向。给出后验均值E（01 y）的精确表示，其中y是来自位置模型f（x - 0）的1 x n观测向量，并且0具有先验密度p（0），即正态尺度混合。令L（0）表示似然函数，令y =（0，a），其中0是最大似然估计，a是最大辅助估计。该表示使用了两个结果：最大似然估计的条件分布p（00，a）（Barndorff-Nielsen，1983）和Masreliez（1975）的结果。结果表明，在正常先验下，E（Ojy）可以表示为p（01 a）的得分函数的线性变换，其中p（01 a）= p（10，a）p（0）d 0。该表示可以被看作是Masreliez结果的推广，该结果处理模型X = 0 +，0 - N（m，r2），并表示X的边缘密度对数的导数的后验均值。附录给出了Masreliez定理所需的一个充分正则性条件和导数与积分符号互换的严格证明。一个表示也开发的情况下，先验密度是一个正常的规模混合。结果具有定量稳健性吸引力。例如，在任何模型和正态先验下，E（Oy）相对于异常观测的灵敏度为：
SUMMARY An exact representation of the posterior mean is developed for a location model and the class of priors that are normal scale mixtures. The result makes use of the conditional distribution of the maximum likelihood estimator and Masreliez's theorem. Directions for future development are indicated. An exact representation for the posterior mean, E(01y), is given where y is a 1 x n vector of observations from a location model, f(x - 0), and 0 has a prior density, p(0), that is a normal scale mixture. Let L(0) denote the likelihood function and let y = (0, a) where 0 is the maximum likelihood estimator and a is the maximal ancillary. The representation makes use of two results: the conditional distribution of the maximum likelihood estimator, p( 00, a) (Barndorff-Nielsen, 1983), and a result of Masreliez (1975). It is shown that, under a normal prior, E(Ojy) can be represented as a linear transformation of the score function of p (01 a), where p (0la)= p(10, a)p(0) dO. The representation can be viewed as a generalization of Masreliez's result that deals with the model, X = 0 + , 0 - N(m, r2) and represents the posterior mean in terms of the derivative of the logarithm of the marginal density for X. The Appendix gives a sufficient regularity condition and a rigorous proof of an interchange of derivative and integral sign required for Masreliez's theorem. A representation is also developed in the case where the prior density is a normal scale mixture. The results have a quantitative robustness appeal. For example, under any model and a normal prior the sensitivity of E( Oy) with respect to aberrant observations is