Computations and applications of mixtures of dirichlet processes
Lynn Kuo · 1980
This dissertation treats three topics which are all related to the mixtures of Dirichlet processes arising as posterior distributions in a Bayesian context. Computation of Bayes estimates based on mixtures of Dirichlet processes is treated in Chapter I. Empirical Bayes problems are selected to illustrate this computation. Bayes estimators are derived from Dirichlet processes priors for general empirical Bayes problems. These estimators may be written as ratios of two multidimensional integrals, each of which may be decomposed into a weighted average of products of one-dimensional integrals. A Monte Carlo method is proposed to approximate numerator and denominator separately. A prior error bound for numerator and denominator separately and a posterior error bound for the ratio are developed to measure the efficiency of the Monte Carlo method. Two numerical examples are presented in the end of Chapter I. The Bayesian bio-assay design problem is treated in Chapter II. It is assumed that the potency curve is a Dirichlet random distribution with parameter (alpha)(t) = Mt, 0 (LESSTHEQ) t (LESSTHEQ) 1, and that n(,1),...,n(,L) animals are treated at drug levels t(,1),...,t(,L) respectively. We find the optimal design levels t(,1),...,t(,L) which minimize the Bayes risk (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where F(t) is the Bayes estimate of F(t), in the following cases: (i) L = 1 and w(t) arbitrary, (ii) w(t) = t, and two animals are treated, (iii) w(t) = t, and L arbitrary, but with M (--->) 0. These results disprove a conjecture of Antoniak. Linear approximation to the potency curve estimate in bio-assay is treated in Chapter III. A Bayes estimate, restricted to the linear space generated by the samples, is derived for the potency curve with squared error loss. In conclusion, some examples and optimal properties of the linear Bayes estimate are presented.