Bayesian duality and risk analysis on the statistical manifold of exponential family with censored data

Information geometry has been attracted wide attentions in the past few decades. This paper focuses on the Bayesian duality on a statistical manifold derived from the exponential family with data from life tests. Based on life testing data, the statistical manifold is constructed with a new cumulant generating function. The Bregman divergence between two parameter points is studied. The dual coordinate system and dual function are obtained. Then, the dualistic structure on the manifold is discussed. The results show that the maximum likelihood estimate can be obtained by minimizing the Bregman divergence induced from the dual function. The Bayesian analysis and prediction are investigated based on informative and non-informative priors. Consider the gamma distribution as an example, the closed-form representations of the dual coordinate system and dual function are obtained. A real data set is employed to illustrate the methodologies and experimental designs developed in this paper.