CAREER: Default Bayesian Methods for Nonparametric Problems

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DEFAULT BAYESIAN METHODS FOR NONPARAMETRIC PROBLEMSStatistical models for complex data often contain one or moreinfinite-dimensional parameters such as a probability density, aregression function, or the transition density of a Markovprocess. The rapid development of innovative Monte-Carlo schemesin the last decade makes it possible to compute Bayesprocedures in these complex problems. However, because of the highdimensionality, it is seldom possible to completely elicit a priorsubjectively from the available information. What is needed is ageneral strategy for constructing priors for infinite-dimensionalparameters that incorporates available prior information, such assmoothness (differentiability) or shape (monotonicity, convexity,unimodality) of a regression function or density function.Ideally, the constructed prior should be tested in the givenproblem to avoid possible pitfalls in estimation. Large-sampleproperties such as consistency and rate of convergence arewell-respected benchmark test criteria. In this research theinvestigator constructs prior distributions for selectproblems using a default approach, devises suitable algorithms forcomputation of the posterior, develops software for computation,investigates the large sample behavior of the resulting procedures,supports the theory and methods via simulation studies withmoderately large samples, and applies the new methods to severalinteresting data sets. The research provides Bayesianmethodologists with a catalog of priors with known performanceproperties, thereby facilitating the application of Bayes methodsin other models with high-dimensional parameters.Modern statistical models for data in a wide variety ofapplications, 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, regressionfunctions, transition densities of Markov chains, and so on. Successfulanalysis of such data presents challenges not found in theanalysis of finite-parameter models, and requires the developmentof 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 thisresearch the investigator develops the theory, methods, andcomputational algorithms for implementing default Bayesian analyses ofcomplex statistical models depending on infinite-dimensionalparameters. The research is disseminated through the teachingof advanced courses and via the usual scientific channels ofpublications and seminars. The research provides newdata-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-relevantapplication areas.