Bayesian inference and relative belief, theory and applications

本研究涉及开发先验的方法，测量由先验引起的偏差，模型检查，检查先验数据冲突，以及基于被称为相对置信比的证据度量的统计推断方法。******虽然统计方法在科学、技术和社会应用中变得越来越重要，但是，对于什么是统计问题的正确表述，从业人员之间几乎没有达成一致意见，而且，即使对组成部分达成一致意见，对解决办法也可能存在分歧。最近对p值使用的关注等争议为这一点提供了一些证据。考虑到统计学家的角色是在数据变化导致不确定性的情况下提供如何推理的指导，这门学科有必要尽可能多地解决由这种冲突引起的模糊性。******分歧的中心点之一集中在使用先验概率分布，因为这些代表了统计分析中明确的主观输入。这些输入似乎与研究者客观的科学传统背道而驰。这种担心无疑是正确的，但同时它忽略了一个明显的事实，即科学调查总是涉及由科学家的选择所导致的主观方面。这项研究的作用是开发工具，以科学有效的方式处理这种主观性。这将在一定程度上通过为广泛的各种情况发展引出论点来实现，以便根据明确阐述的标准作出选择。此外，将开发方法，允许测量和控制这些选择所带来的偏差，并允许评估所做的选择是否与客观数据相冲突。******一旦选择了统计分析的成分，并且至少部分验证了所描述的，下一步就是将统计推断理论的规则与数据一起应用于这些成分以导出推断。一个相关的推理理论应该基于一个关于如何衡量与感兴趣的问题相关的统计证据的明确定义。对于统计理论的许多方法来说，缺乏对这种证据度量的明确定义又产生了另一种模糊性。相对信念推理理论的发展是为了提供一种基于有效的统计证据度量的方法。本研究计划还涉及相关理论的进一步发展，并将其应用于各种具有实际意义的统计问题。******这项研究还需要开发合适的近似计算工具。*****

This research is concerned with developing methodology for the elicitation of priors, measuring the bias induced by a prior, model checking, checking for prior-data conflict and methods for statistical inference based upon a measure of evidence known as the relative belief ratio. ******While statistical methodology is becoming increasingly important in scientific, technological and social applications, there is little agreement among practitioners as to what constitutes the correct formulation of a statistical problem and furthermore, even when the ingredients are agreed upon, there may be disagreement concerning the solution. Controversies such as the recent concerns over the use of p-values provide some evidence for this. Given that the role of a statistician is to provide guidance on how one is to reason in contexts where data variation results in uncertainty, it is necessary for the discipline to do as much as possible to resolve the ambiguities arising from such conflicts.******One of the central points of disagreement focuses on the use of prior probability distributions as these represent clear subjective inputs into statistical analyses. These inputs would seem to run contrary to the scientific tradition of the objectivity of the investigator. This concern is undoubtedly valid, but at the same it ignores the obvious fact that scientific investigations always involve subjective aspects resulting from the choices made by the scientist. The role of this research is to develop tools for dealing with this subjectivity in a scientifically valid fashion. This will be accomplished in part through the development of elicitation arguments for a broad variety of contexts so that the choices are based on clearly articulated criteria. Furthermore, methodology will be developed that allows for the measurement and control over the biases introduced by these choices and allows for the assessment of whether or not the choices made are in conflict with the objective data.******Once the ingredients of a statistical analysis have been chosen, and at least partially validated as described, the next step is to apply the rules of a theory of statistical inference to these ingredients together with the data to derive inferences. A relevant theory of inference should be based on a clear definition of how to measure the statistical evidence relevant to questions of interest. For many approaches to statistical theory, the lack of a clear definition of such a measure of evidence produces yet another ambiguity. The theory of relative belief inference has been developed to provide an approach that is based on a valid measure of statistical evidence. This research proposal is also concerned with the further development of the associated theory and the application of this to a broad variety of statistical problems with practical relevance. ******This research will also require the development of suitable computational tools for approximations.*****