Intrinsic losses
Intrinsic losses
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DOI：
10.1007/bf00133173
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发表时间：
1996-03-01
影响因子：
0.8
通讯作者：
Robert, CP
中科院分区：
文献类型：
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作者：
Robert, CP
由于特定损失函数的选择对结果的推断有很大影响，当没有关于决策者效用函数的信息时，似乎有必要依赖“内在”损失，而不是要求使用经典损失，如平方误差损失。由于这种设置与贝叶斯分析中非信息先验的推导非常相似，我们首先回顾一下推导的条件，并从这些条件中推导出对固有损失的一些要求。这样看来，这些损失函数应该只依赖于抽样分布，并且它们应该独立于分布的参数。因此，由此得到的估计量是变换等变的。我们研究了两种自然固有损失，即熵和Hellinger损失的性质，并证明了它们可以用指数族的封闭形式来表示。此外，熵损失还提供了共轭先验下贝叶斯估计的解析表达式；与Hellinger损失相关的贝叶斯估计的推导更加繁琐，如Poisson和Gamma情形所示，同时导致类似的估计。
Since the choice of a particular loss function strongly influences the resulting inference, it seems necessary to rely on ''intrinsic'' losses when no information is available about the utility function of the decision-maker, rather than to call for classical losses like the squared error loss. Since this setting is quite similar to the derivation of noninformative priors in Bayesian analysis, we first recall the conditions of this derivation and deduce from these conditions some requirements on the intrinsic losses. It then appears that these loss functions should only depend on the sampling distribution and that they should be independent of the parameterization of the distribution. The resulting estimators are therefore transformation equivariant. We study the properties of two natural intrinsic losses, namely entropy and Hellinger losses, and show that they can be expressed in closed form for exponential families. Moreover, the entropy loss also provides analytic expressions of Bayes estimators under conjugate priors; the derivation of Bayes estimators associated with the Hellinger loss is more cumbersome, as shown in Poisson and Gamma cases, while leading to similar estimators.