Mathematical Sciences: Assessing Robustness of Inference

所有的统计程序都会产生一种推论，一种基于观察数据和假设模型的陈述。由于通常存在难以(或不可能)检验的模型假设，因此需要针对模型假设进行稳健的推论。我们最关心的推论是POST(数据精度度量)置信集和假设检验。这样的度量通常是通过决策理论或贝叶斯论证得出的，因此是基于关于抽样分布、损失函数和先验分布的假设。在这里，我们希望研究精度估计器的性能，使用频率标准和贝叶斯标准，因为基本的假设是放松的。我们最感兴趣的是从默认先验构建的过程，因为这些过程往往在频率和贝叶斯审查下都表现良好。我们将解决抽样稳健性的一种方法是将默认先验与经验似然性相结合，以获得(某种程度上)自动推理稳健性过程。推理的稳健性将使用各种标准来判断，例如后验概率的范围，多重和距离惩罚损失函数，以及新的(和有希望的)概率膨胀理论。

All statistical procedures result in an inference, a statement that is based on both the observed data and the assumed model. Since there are often model assumptions that are difficult (or impossible) to check, it is desirable for inferences to be robust against model assumptions. The inferences that we are most concerned about are post (data accuracy measures for confidence sets and hypothesis tests. Such measures are often derived through decision theoretic or Bayesian arguments, hence are based on assumptions about sampling distributions, loss functions, and prior distributions. Here we want to investigate the performance of accuracy estimators, using both frequentist and Bayesian criteria, as the underlying assumptions are relaxed. We are most interested in procedures constructed from default priors, for these tend to perform well under both frequentist and Bayesian scrutiny. One way that we will address the sampling robustness is to combine the default priors with an empirical likelihood to obtain a (somewhat) automatic inference robust procedure. Robustness of inference will be judged using a variety of criteria such as ranges of posterior probabilities, multiple and distance penalizing loss functions and a new (and promising) theory of dilation of probabilities.