Post Model Selection Inference and Empirical Bayes Methods

考虑一个包含p个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只包含这些可能预测因子的一个子集。如果协变量是相关的，基于所选模型的常规推断可能无效；例如，置信区间覆盖所选模型的真实参数值的概率可能被严重夸大。研究人员提出了一个版本的经典推理标准和相应的方法，以保证后选择推理将在这些标准内有效。推断是保守的，因为它独立于所使用的模型选择方法是有效的，并且对于所有参数配置保证了正确的（尽管可能是保守的）边际覆盖。这个过程在算法上很容易描述。然而，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p值的这些概率的可行计算仍在研究中。尽管如此，我们还是可以推导出一些有用的渐近界，并且可以更精确地分析一些重要的特殊情况。传统的统计推断要求在分析数据之前知道数据是如何产生的模型。然而，在涉及诸如方差分析和多元回归等常见程序的应用程序中，通常首先进行一个或多个模型选择程序，以帮助确定分析的模型。然后，在选择模型之后进行统计检验和计算置信区间，就好像在检查数据之前已经选择了最终模型一样。这样的例子在社会科学、计量经济学文献、流行病学和基因组学中比比皆是。本建议首先审查这种做法的后果，以便对其可能造成误导和误导的程度进行分类。如果没有额外的注意，被估计的参数就不再被很好地定义，并且模型选择后的抽样分布具有与没有模型选择的情况非常不同的属性。统计推断，如置信区间和统计检验，并不像通常假设的那样执行。许多作者已经注意到一些或所有这些问题，但没有提出有效的一般统计推断程序来处理这种情况。研究者提出并研究了一种方法，该方法可以根据观察到的数据在选择的模型中产生有效的统计推断。所提出的方法是普遍有效的，独立于用于选择要保留在模型中的变量的过程。因此，从这个角度来看，没有必要研究当前使用的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提高了我们的置信区间建议的性能，这方面的一些方面自然会包括在我们的研究中。特别是基于非参数贝叶斯思想的一些新的模型选择方法，将从灵活地产生令人满意的模型的能力和从模型后选择推理的角度进行研究。这些后模型选择思想的扩展也将在各种统计设置中进行探索，而不是最常见的高斯线性模型，这是本提案的初始目标。

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.