Differentiable Statistical Functionals and Bayes Asymptotics

达德利9704603 对于经验分布函数上泛函或算子的微分，大多数统计学家采用了紧可微性或Hadamard可微性。R.达德利和他的同事证明了弗雷歇 可微性，众所周知，当它成立时是上级的，可以通过使用一维样本空间的p-变差范数来获得，其中最佳界是关于可微性的。该计划下一步将在多维空间中进行。因此，R。达德利和他的一位同事证明，当概率很小时，即使相对误差很小，后验概率也可以很好地近似于正态（高斯）概率。 在可微统计泛函的研究中，人们正在研究估计量或其他统计量如何依赖于观察到的数据分布：这种依赖是否平滑，近似线性，以及如何量化平滑或缺乏平滑。在贝叶斯渐近中，人们从可能参数值的假设表面上的假设先验概率分布开始，并从足够的观察中找到近似高斯形式的后验分布。该项目表明，一个简单的方法工作得很好，给出了非常准确的高斯近似的后验概率。 通过这些近似，该项目包括基于多个数据集在竞争统计模型或理论之间进行选择的方法研究，如在相同或密切相关问题的多个实验研究中。

Dudley 9704603 For differentiating functionals or operators on empirical distribution functions, most statisticians had adopted compact or Hadamard differentiability. R. Dudley and co-workers are showing that Frechet differentiability, well known to be superior when it holds, can be obtained by use of p-variation norms for one-dimensional sample spaces, with optimal bounds on remainders. The program is next being pursued in multidimensional spaces. Also, R. Dudley and a co-worker are showing that a posteriori probabilities can be well approximated by normal (Gaussian) probabilities even with respect to relative error when the probabilities are very small. In the study of differentiable statistical functionals, one is looking at how estimators or other statistics depend on the observed data distribution: is the dependence smooth, approximately linear, or not, and how can smoothness or lack thereof be quantified. In Bayes asymptotics, one begins with an assumed prior probability distribution over an assumed surface of possible parameter values, and from enough observations, finds an a posteriori distribution which has approximately a Gaussian form. The project is showing that a simple method works very well, giving very accurate Gaussian approximations to the a posteriori probabilities. By way of these approximations, the project includes studies of methods for choosing between competing statistical models, or theories, based on multiple data sets, as in multiple experimental studies on the same or closely related questions.