A fundamental class of inferential problems are those characterised by there having been a substantial degree of pre-data (or prior) belief that the value of a model parameter $\theta_j$ was equal or lay close to a specified value $\theta^{*}_j$, which may, for example, be the value that indicates the absence of a treatment effect or the lack of correlation between two variables. This paper puts forward a generally applicable 'push-button' solution to problems of this type that circumvents the severe difficulties that arise when attempting to apply standard methods of inference, including the Bayesian method, to such problems. Usually the only input of major note that is required from the user in implementing this solution is the assignment of a pre-data or prior probability to the hypothesis that the parameter $\theta_j$ lies in a narrow interval $[\theta_{j0},\theta_{j1}]$ that is assumed to contain the value of interest $\theta^{*}_j$. On the other hand, the end result that is achieved by applying this method is, conveniently, a joint post-data distribution over all the parameters $\theta_1,\theta_2,\ldots,\theta_k$ of the model concerned. The proposed method is constructed by naturally combining a simple Bayesian argument with an approach to inference called organic fiducial inference that was developed in a number of earlier papers. To begin with, the main theoretical arguments underlying this combined Bayesian and fiducial method are presented and discussed in detail. Various applications and useful extensions of this methodology are then outlined in the latter part of the paper. The examples that are considered are made relevant to the analysis of clinical trial data where appropriate.
翻译:一种基本的推论问题, 其特征是, 模型参数 $\theta_ j$ 的值等于或接近指定值$\theta_ j$, 例如, 可能是表示不存在处理效果或两个变量之间缺乏相关性的值。 本文针对这类问题提出了一个普遍适用的“ push- button ” 解决方案, 这些问题回避了在试图对此类问题采用标准推算方法( 包括贝耶斯方法) 时出现的严重困难。 通常, 用户在执行此解决方案时需要的主要注释的输入值是等于或接近指定值$\ theta_ j$, 例如, 表示没有处理效果或两个变量之间缺乏相关性。 本文提出了一种普遍适用的“ push- button ” 解决方案。 在另一个方面, 应用这一方法的最终结果是, 方便地, 将先前的推算值与当前推算值的理论值合并起来 。 此推算中, 此推算的推算法的推理推算法的推算法是 。 此推算法中, 此推算推算的推算法的推算法的推算中, 此推算法的推算推算法的推算中, 的推算法推算法推算法推算的推算法是一个推算法的推算法推算法推算法的推算法的推算法的推算法中, 。