Differentiable Statistical Functionals and Bayes Asymptotics

研究者和同事正在研究经验测度和分布函数的泛函的可微性。在一维中，p变模对Frechet可微性很有效。正在寻求向几个方面进行扩展。在贝叶斯渐近中，即使中间偏差的概率很小，也可以发现具有较小相对误差的正态近似。在贝叶斯渐近中，目标之一是选择几个统计模型中最好的一个，可能适用于多个数据集。例如，可以对一种疾病的治疗方法进行多次临床试验。三种模式是治疗是有益的，有害的，或者没有区别。进一步的模型考虑了治疗在不同的研究人群中可能有本质上不同的效果的可能性。该过程是从每个模型上的非信息先验概率分布开始，然后根据每个数据集的可能性对其进行调整。正在研究更新后的概率的改进近似值。对于可微的统计函数，数据集给出了概率分布的近似值。如果一个非线性变换同时应用于真分布函数和它的近似值，人们就会在真分布的邻域中寻找一个尽可能接近非线性的线性变换。人们还在寻找有效的方法来限制真实分布和近似分布及其变换之间的差异。

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.