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Robust Bayesian analysis
Abstract
In Chapter 2, the robustness of Bayes analysis with reference to conjugate prior classes is discussed. The optimal robust posterior confidence set is studied for the parameter $\theta$ when the prior varies over a class $\Gamma$. For the one-dimensional normal-normal case, a convenient formula and tables can be used to find the optimal robust $\Gamma$-posterior confidence set. For the Poisson-gamma and the exponential-gamma case, an one dimensional optimization procedure is given for the two dimensional optimization problem to find the optimal robust $\Gamma$-posterior confidence set. The two dimensional optimal robust set for normal-normal case is discussed. In Chapter 2, the $\epsilon$-contamination class is extended to a class in which $\pi\sb0$ belongs to a conjugate class. Several results of Berger and Berliner (1986) are extended. The ML-II prior and the optimal robust $\Gamma$-posterior confidence set for the $\epsilon$-contamination classes and extended $\epsilon$-contamination classes are studied. Relations among the length and the center of the optimal robust interval, the contamination $\epsilon$ and the variance of the prior for normal-normal cases are studied. When the hierarchical prior class is too wide, the prior which is most unlikely may adversely affect the robustness, due to long tails of the hierarchical prior. Robustness is improved if the hierarchical prior is trimmed. Entropy prior has been proposed in literature as a noninformative prior when some information about the prior is available. In Chapter 3, we use other diversity measures such as Gini-Simpson type divergence, Lorenz curve divergence and Fisher-information to find the prior under some given conditions which may contain the given percentiles. It is difficult from entropy, the solution is continuous for the Fisher information, even when the restraints involve percentiles.
Subject Area
Statistics
Recommended Citation
Li, Yuanzhang, "Robust Bayesian analysis" (1990). ETD collection for University of Nebraska-Lincoln. AAI9118466.
https://digitalcommons.unl.edu/dissertations/AAI9118466