Collaborative Research: Generalized Fiducial Inference - An Emerging View

这一建议的动机是广义基准推理的成功，作为费舍尔基准论点的推广，PIS引入了广义基准推理。由于私人投资机构对广义基准法的理论和应用进行了许多研究，可以得出以下重要结论：(A)对于大类问题，有一个统一和系统的程序来开发基准解；(B)基准法通常导致非常有效的推理程序，因此它们与使用其他方法开发的程序具有竞争力；(C)基准法在大类问题中是渐近正确的；(D)通过适当选择先验，许多基准点分布也可以实现为贝叶斯后验。(E)贝叶斯方法和基准方法都产生了有用的区间推理程序，正如在这两个领域的各种出版物中所确立的那样。显然，任何一种方法都不能声称支配另一种方法；(F)两种方法通常都需要对所需后验分布或基准分布的实际数值计算进行MCMC模拟。在适当考虑了基准法有望在理论上和实践上产生新的有用结果的统计推断领域之后，投资促进机构建议对以下专题进行研究：(A)广义线性混合模型的区间数据基准框架的扩展以及相关的计算方法；(B)在广义基准推理框架内扩展投资促进机构的工作以解决模型选择问题；(C)对参数的罗伯特化基准分布的概念进行定义和研究，并开发从数据计算该分布的计算方法；(D)在一些标准的参数例子中应用稳健的基准方法来得到新的稳健的推理方法；(E)发展一些通用的计算策略来实现复杂的实际问题的基准方法。这项工作的影响将对公共政策产生直接影响。例如，美国食品和药物管理局(FDA)的指导文件详细说明了证明两种或两种以上药物配方等效性的分析程序。PIs旨在表明，基准方法将导致更有效的程序，这将导致成本和时间的节省，这对制药业来说是一个重要的问题。在计量方面，国际度量局(BIPM)与国际标准化组织(ISO)联合出版了一份《测量不确定度表达指南》(GUM)，其中给出了美国NIST、英国NPL和德国PTB等国家计量机构应遵循的程序。计量学特有的一个问题是，每一次测量都会受到未知和不可知的系统误差的影响，这些误差往往比随机误差更大。目前唯一已知的量化这些不可知的系统误差的方法是通过指定它们的主观分布。GUM规定了一些特别的方法，用于结合基于数据的某些误差分量的标准偏差估计和其他误差分量的主观不确定度估计。PI旨在证明，基准法为实现这一目标提供了一种新的自然方法。这样的结果可能会影响计量界修改和改进他们目前的程序。

This proposal is motivated by the success of Generalized Fiducial Inference as introduced by the PIs as a generalization of Fisher's fiducial argument. As a result of the many studies conducted by the PIs on the theory and applications of generalized fiducial methods the following important conclusions can be made: (a) A unified and systematic procedure is available for developing fiducial solutions for large classes of problems; (b) The fiducial approach generally leads to very efficient inference procedures and thus they are competitive with procedures developed using other approaches; (c) Fiducial procedures are asymptotically correct in large classes of problems; (d) Many fiducial distributions can also be realized as a Bayesian posterior by an appropriate choice of a prior. However, this is not always possible, which establishes that the two approaches are not equivalent in general; (e) Both the Bayesian approach and the fiducial approach lead to useful interval inference procedures as have been established in various publications in both areas. It is clear that neither approach can claim to dominate the other; (f) Both approaches typically require MCMC simulations in regards to actual numerical computation of the required posterior or fiducial distributions. After giving due consideration to areas of statistical inference where a fiducial approach is expected to lead to new and useful results, both theoretical and practical, the PIs propose to conduct research into the following topics: (a) Extensions of Interval Data fiducial framework for Generalized Linear Mixed Models together with associated computational approaches; (b) Extension of the work of the PIs to address the model selection problem within the Generalized Fiducial Inference framework; (c) Definition and investigation of the concept of a Robustified Fiducial Distribution for a parameter and development of computational methods for calculating it from data; (d) Application of robust fiducial approaches to arrive at new robust inference methods in some standard parametric examples; (e) Development of some general computational strategies for implementation of fiducial methods for complex practical problems.This proposal studies a new approach to statistical inference based on Fisher's fiducial argument. The implications of this work will have an immediate effect on public policy. For instance, the U.S. Food and Drug Administration (FDA) guidance document spells out analysis procedures for demonstration of equivalence of two or more drug formulations. The PIs aim to show that the fiducial approach will lead to more efficient procedures, which will result in cost and time savings, an important issue for the drug industry. In metrology, the International Bureau of Weights and Measures (BIPM) in conjunction with the International Organization for Standardization (ISO), has published a ``Guide to Expression of Uncertainty in Measurements'' (GUM) which gives the procedures to be followed by national metrological institutes such as NIST in the US, NPL in UK, and PTB in Germany. A problem that is unique to metrology is that every measurement is subject to unknown and unknowable systematic errors that are often larger than random errors. The only currently known way to quantify these unknowable systematic errors is via specification of subjective distributions for them. The GUM specifies some ad hoc methods for combining data-based estimates of standard deviations for some error components and subjective estimates of uncertainty for other error components. The PIs aim to demonstrate that the fiducial method provides a new natural approach for accomplishing this. Such results are likely to influence the metrology community in modifying and improving their current procedures.