Differentiable Statistical Functionals and Bayes Asymptotics

Dudley 9704603对于经验分布函数上的泛函或算子的微分，大多数统计学家都采用了紧致性或Hadamard可微性。R.Dudley和他的同事们证明了Frechet可微性，众所周知，当它成立时，它是优越的，可以通过使用一维样本空间的p-变分范数来获得，并且具有关于剩余项的最优界。该计划下一步将在多维空间进行。此外，R.Dudley和他的一位同事正在证明，即使在概率非常小的情况下，后验概率也可以很好地用正态(高斯)概率来逼近，即使相对于相对误差也是如此。在可微统计泛函的研究中，人们关注的是估计量或其他统计量如何依赖于观察到的数据分布：这种相关性是光滑的、近似线性的还是不是的，以及如何对其平滑或缺失进行量化。在贝叶斯渐近论中，人们从假设的可能参数值的假设表面上的先验概率分布开始，并从足够的观测中找到具有近似高斯形式的后验分布。该项目表明，一种简单的方法非常有效，给出了非常准确的后验概率的高斯近似。通过这些近似，该项目包括研究在基于多个数据集的相互竞争的统计模型或理论之间进行选择的方法，例如在关于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.