This thesis investigates the problems of discimination and regression using Bayesian methods with emphasis on their asymptotic properties when p, the number of variables that can be observed, is unlimited. For the problem of discriminating between two multivariate normal populations the conjugate prior is found to lead to asymptotically perfect discrimination, under certain conditions on the parameters. Similarly, in a problem of discrimination between two populations with binary variables, using a Dirichlet process prior, necessary and sufficient conditions for asymptotically perfect discrimination are found. To investigate this determinism a comparison is made between the Bayesian discriminant function and a sample-based discriminant function which fits the data exactly when p is large. It is shown that their performances are asymptotically equivalent. Similarly, for the regression of normal variables with a conjugate prior the Bayes predictor, which implies asymptotic deterministic predictability, is asymptotically equivalent to a classical least squares predictor which exacly fits the sample data for large p. Thus the conjugate Bayesian approach neglects the problem of bias due to overfitting. In contrast, it is shown that a certain nonconjugate prior does not imply asymptotic determinism for the Bayes predictor, and renders the behaviours of Bayes and least squares predictors different. This reveals the importance of the choice of prior distribution for Bayesian analysis.
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