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.
