CAREER: Default Bayesian Methods for Nonparametric Problems

North Carolina State University, Raleigh NC

Investigators

Abstract

DEFAULT BAYESIAN METHODS FOR NONPARAMETRIC PROBLEMS Statistical models for complex data often contain one or more infinite-dimensional parameters such as a probability density, a regression function, or the transition density of a Markov process. The rapid development of innovative Monte-Carlo schemes in the last decade makes it possible to compute Bayes procedures in these complex problems. However, because of the high dimensionality, it is seldom possible to completely elicit a prior subjectively from the available information. What is needed is a general strategy for constructing priors for infinite-dimensional parameters that incorporates available prior information, such as smoothness (differentiability) or shape (monotonicity, convexity, unimodality) of a regression function or density function. Ideally, the constructed prior should be tested in the given problem to avoid possible pitfalls in estimation. Large-sample properties such as consistency and rate of convergence are well-respected benchmark test criteria. In this research the investigator constructs prior distributions for select problems using a default approach, devises suitable algorithms for computation of the posterior, develops software for computation, investigates the large sample behavior of the resulting procedures, supports the theory and methods via simulation studies with moderately large samples, and applies the new methods to several interesting data sets. The research provides Bayesian methodologists with a catalog of priors with known performance properties, thereby facilitating the application of Bayes methods in other models with high-dimensional parameters. Modern statistical models for data in a wide variety of applications, such as data mining, image analysis, biometrics, biostatistics, bioinformatics, signal processing, and finance, often depend on high- or infinite-dimensional parameters such as survival distributions, probability densities, regression functions, transition densities of Markov chains, and so on. Successful analysis of such data presents challenges not found in the analysis of finite-parameter models, and requires the development of new statistical theory, methods and software. A non-subjective Bayesian method retains the advantages of the Bayesian paradigm without requiring a subjective prior elicitation. In this research the investigator develops the theory, methods, and computational algorithms for implementing default Bayesian analyses of complex statistical models depending on infinite-dimensional parameters. The research is disseminated through the teaching of advanced courses and via the usual scientific channels of publications and seminars. The research provides new data-analytic tools for solving problems arising in diverse fields. Useful priors with known performance are cataloged and user friendly software is developed for ready applications to diverse fields. Thus the research has a major impact on the conduct of science in a number of highly-relevant application areas.