Post Model Selection Inference and Empirical Bayes Methods

模型选择后推理和经验贝叶斯方法

• 批准号: 1007657
• 负责人: Lawrence Brown
• 金额: $ 40万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007657&HistoricalAwards=false
• 关键词: Post; Model; Selection; Inference; Empirical

Consider a standard Gaussian multiple regression model involving p independent covariates. In many applications a first step of the analysis is to reduce the data via model selection to one containing only a subset of these possible predictors. If the covariates are correlated, conventional inference based on the selected model may be invalid; for example, probabilities that confidence intervals cover the true parameter values for the selected model may be grossly overstated. The investigators propose a version of classical inference criteria and a corresponding method for guaranteeing that post selection inferences will be valid within these criteria. The inference is conservative in that it is valid independent of the model selection method that was used, and correct (though possibly conservative) marginal coverage is guaranteed for all parameter configurations. The procedure is algorithmically easy to describe. However in its optimal implementation requires numerical estimation of certain probabilities related to high dimensional Gaussian distributions, and feasible computation of these probabilities for larger values of p is an issue still under investigation. Notwithstanding certain useful asymptotic bounds can be derived, and some important special cases can be analyzed with greater precision. Conventional statistical inference requires that a model of how the data were generated be known before the data are analyzed. Yet in applications involving such common procedures as the Analysis of Variance and multiple regression it is often the case that one or more model selection procedures are first undertaken in order to help determine a model for the analysis. This model selection is then followed by statistical tests and confidence intervals computed as if the final model had been chosen in advance of examining the data. Examples abound in the social sciences, in the econometric literature, in epidemiology and in genomics. This proposal begins by examining consequences of such a practice in order to categorize the degree to which it may be misleading and misguided. Without additional care the parameters being estimated are no longer well defined, and post-model-selection sampling distributions have properties that are very different from what would be the case without model selection. Statistical inference such as confidence intervals and statistical tests does not perform as is customarily assumed. Many authors have noted some or all of these problems, but have not proposed valid general statistical inference procedures to cope with the situation. The investigators propose and study a method that produces valid statistical inference within the models selected based on the observed data. The proposed approach is universally valid, independent of the procedure that was used to select the variables to be retained in the model. Thus, from this perspective it is not necessary to investigate the details of the various model selection proposals in current use. Nevertheless, certain models and model selection procedures do yield improved performance of our confidence interval proposal, and some aspects of this will naturally be included in our research. In particular some new model selection methods based on nonparametric Bayesian ideas will be investigated both for their ability to flexibly produce satisfactory models and from the perspective of post model selection inference. Extension of these post model selection ideas will also be explored in a variety of statistical settings beyond the most common Gaussian linear models that are the initial target of this proposal.

考虑一个涉及P独立协变量的标准高斯多元回归模型。在许多应用程序中，分析的第一步是通过模型选择将数据减少到仅包含这些可能预测因子子集的数据。如果协变量是相关的，则基于所选模型的常规推断可能无效；例如，置信区间涵盖所选模型的真实参数值的概率可能会被严重夸大。调查人员提出了一种经典推理标准的版本，并提出了一种确保选择后推断的相应方法在这些标准中有效。该推论是保守的，因为它与所使用的模型选择方法无关，并且可以保证所有参数配置（尽管可能是保守的）边缘覆盖范围。该过程在算法上很容易描述。但是，在其最佳实现中，需要对与高维高斯分布相关的某些概率进行数值估计，并且对于较大的P值，对这些概率的可行计算仍在研究中。尽管可以得出某些有用的渐近界限，并且可以更精确地分析一些重要的特殊情况。 常规的统计推断要求在分析数据之前知道如何生成数据的模型。然而，在涉及诸如方差分析和多元回归之类的常见程序的应用中，通常首先采用一个或多个模型选择程序，以帮助确定分析模型。然后，此模型选择之后是统计测试和置信区间计算的，就好像在检查数据之前已选择了最终模型一样。社会科学，计量经济学文献，流行病学和基因组学中的例子比比皆是。该建议首先要检查这种做法的后果，以便对可能具有误导性和误导性的程度进行分类。如果不额外护理，估计的参数不再定义，并且模型选择后的采样分布具有与没有模型选择的情况大不相同的属性。统计推断（例如置信区间和统计检验）并未按照通常的假设执行。许多作者注意到了一些或全部这些问题，但并未提出有效的一般统计推理程序来应对情况。研究人员提出并研究一种基于观察到的数据选择的模型中产生有效统计推断的方法。所提出的方法是普遍有效的，与选择要保留在模型中的变量的过程无关。因此，从这个角度来看，无需研究当前使用中各种模型选择建议的细节。然而，某些模型和模型选择程序确实会提高我们的置信区间建议的性能，并且其中的某些方面自然会包括在我们的研究中。特别是，将研究一些基于非参数贝叶斯想法的新模型选择方法，既将研究它们灵活地产生令人满意的模型的能力，也将从后选择推理的角度进行研究。这些邮政模型选择思想的扩展也将在最常见的高斯线性模型之外的各种统计环境中进行探讨，这是该提案的初始目标。