Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference
Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference
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DOI：
10.1080/01621459.2012.747960
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发表时间：
2013-03-01
影响因子：
3.7
通讯作者：
Liu, Chuanhai
中科院分区：
文献类型：
--
作者：
Martin, Ryan;Liu, Chuanhai
没有先验的后验概率统计推断是一个重要但至今难以实现的目标。Fisher的置信推理，Dempster-Shafer理论的信念函数，贝叶斯推理与默认先验是试图实现这一目标，但迄今为止，没有一个给出了一个完全令人满意的图片。本文提出了一种新的概率推理框架，推理模型（IM）的基础上，它不仅提供了数据相关的概率测量的未知参数的不确定性，但也这样做的自动长期运行的频率校准属性。这种新方法的关键是识别与可观测数据和未知参数相关联的不可观测辅助变量，并在对数据进行调节之前用随机集预测该辅助变量。在这里，我们提出了一个三步IM建设，并证明了IM的信度函数在温和的条件下的频率校准属性。相应的最优性理论的发展，这有助于解决非唯一性问题。几个例子来说明这种新的方法。
Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This article presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but also does so with an automatic long-run frequency-calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM's belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the nonuniqueness issue. Several examples are presented to illustrate this new approach.