The conditionality principle in high-dimensional regression
高维回归中的条件性原则AI 解读 ↓
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高维回归中的条件性原则
- 研究问题
- 当协变量数超过样本量时，对协变量条件化如何影响响应条件方差的可估计性，协变量边际信息又有何作用？
- 方法与思路
- 作者比较条件与无条件框架，利用条件框架下贝叶斯检验的风险论证估计限制，并研究正态协变量下的条件误差。
- 材料陈述的贡献
- 摘要称，条件框架存在一致估计的不可能性，但已知正态协变量分布可使相关估计器的条件误差以高概率趋零。
- 阅读关注点
- 应核对一致性的定义、参数空间与贝叶斯风险论证，以及条件不可能性如何与高概率条件误差收敛并存。
- 阅读边界
- 仅据 Crossref 摘要；具体参数空间与收敛条件未提供，不能将条件框架的不可能性解释为任何高维方差估计都无效。
原文摘要
Summary Consider a high-dimensional linear regression problem, where the number of covariates is larger than the number of observations and the interest is in estimating the conditional variance of the response variable given the covariates. A conditional and an unconditional framework are considered, where conditioning is with respect to the covariates, which are ancillary to the parameter of interest. In recent papers, a consistent estimator was developed in the unconditional framework when the marginal distribution of the covariates is normal with known mean and variance. In the present work, a certain Bayesian hypothesis test is formulated under the conditional framework, and it is shown that the Bayes risk is a constant. This implies that no consistent estimator exists in the conditional framework. However, when the marginal distribution of the covariates is normal, the conditional error of the above consistent estimator converges to zero, with probability converging to one. It follows that even in the conditional setting, information about the marginal distribution of an ancillary statistic may have a significant impact on statistical inference. The practical implication in the context of high-dimensional regression models is that additional observations where only the covariates are given are potentially very useful and should not be ignored. This finding is most relevant to semi-supervised learning problems where covariate information is easy to obtain.