"Nonlinear Bayesian Function Estimation in Complex Models"

本研究的基本主题是非参数贝叶斯方法的最新进展，如混合模型和Dirichlet过程模型，以及一些重要的统计模型和概念，包括非线性自回归，神经网络和小波表示的合成。在拟议的研究的关键要素是计算问题。 具体的问题包括马尔可夫链蒙特卡罗模拟估计混合模型，不允许直接吉布斯采样的实施，估计可变结构的神经网络，估计后验分布的小波表示的一个适当的概率模型的系数。神经网络研究将基于将神经网络方法制定为分层非线性回归模型，该模型混合了随机数量的项和层次。小波表示的工作将提供阈值规则作为分层贝叶斯模型中的最大后验估计。 这个项目将贝叶斯函数估计的方法与应用领域中出现的问题结合起来。特别是，研究涉及神经网络模型，小波表示和非线性自回归。 神经网络的应用在生物学、心理学、物理学、工程学和计算机科学中都有。小波表示已被发现在清除噪声数据中是有用的。 非线性自回归是研究时间序列数据的广泛使用的方法的扩展。

The underlying theme of this research is the synthesis of recent advances in nonparametric Bayesian methods, such as mixture models and Dirichlet process models, and some important statistical models and concepts, including nonlinear autoregression, neural networks and wavelet representations. Key elements in the proposed research are computational issues. Specific problems include Markov chain Monte Carlo simulations to estimate mixture models which do not allow straightforward Gibbs sampling implementation, estimation of variable architecture neural networks, and estimation of the posterior distribution of the coefficients of a wavelet representation of an appropriate probability model. The neural network research will be based on formulating the neural network approach as a hierarchical nonlinear regression model with a mixture involving a random number of terms and hierarchies. Work on the wavelet representations will provide thresholding rules as maximum posterior estimates in a hierarchical Bayes model. This project brings together methodology from Bayesian function estimation with problems arising in applied fields. In particular, the research deals with neural network models, wavelet representations, and nonlinear autoregression. Applications of neural networks are found in biology, psychology, physics, engineering, and computer science. Wavelet representations have been found useful in cleaning noisy data. Nonlinear autoregression is an extension of widely used methods to study time series data.