Differentiable Statistical Functionals and Bayes Asymptotics

研究人员和同事正在研究经验度量泛函和分布函数的可微性。在一维空间中，p-变分范数对Frechet可微性有很好的效果。目前正在寻求对几个方面的扩展。在贝叶斯渐近论中，即使对于小的中间偏差概率，也能找到相对误差很小的正态近似。在贝叶斯渐近论中，目标之一是从几个统计模型中选择最好的，可能是针对多个数据集。例如，可以对一种疾病的治疗进行多个临床试验。三种模式是，治疗是有帮助的，是有害的，或者没有区别。进一步的模型包含了这样一种可能性，即治疗在不同的研究人群中可能具有显著不同的效果。这个过程是从每个模型的无信息的先验概率分布开始，然后根据每个数据集的可能性进行调整。正在研究更新概率的改进近似值。对于可微统计泛函，数据集给出了概率分布的近似值。如果将非线性变换既应用于真实分布函数，又应用于其近似，则可以在真实分布的邻域内寻找近似于非线性函数的线性变换。人们还在寻找有效的方法来限制真实分布和近似分布之间的差异及其变换。

The investigator and co-workers are studying differentiability of functionals of empirical measures and distribution functions. In one dimension, p-variation norms work well for Frechet differentiability. Extensions to several dimensions are being pursued. In Bayes asymptotics, normal approximations with small relative errors are being found even for small probabilities of intermediate deviations.In Bayes asymptotics, one of the goals is to choose the best of several statistical models, possibly for multiple data sets. For example, multiple clinical trials may be done of a treatment for a disease. Three models are that the treatment is helpful, is harmful, or makes no difference. Further models incorporate the possibility that the treatment may have substantially different effects in different study populations. The procedure is to begin with a noninformative prior probability distribution on each model, then adjust it based on the likelihoods from each data set. Improved approximations of the updated probabilities are being investigated. For differentiable statistical functionals, a data set gives an approximation to a probability distribution. If a nonlinear transformation is applied both to the true distribution function and to its approximation, one looks for a linear transformation that approximates the nonlinear one as well as possible in the neighborhood of the true distribution. One is also looking for effective ways of bounding the discrepancy between the true and approximate distributions and their transformations.