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The classical Ellsberg experiment provides decision-makers with a choice problem in which the likelihood of winning the gambling is uncertain. In this paper, we consider a two-dimensional Ellsberg experiment in which the likelihood of winning the gambling and the income after that are both uncertain, establish a mathematical model to characterize the corresponding Ellsberg experiment and solve the choice problem based on probability theory, uncertainty theory, and chance theory. [ABSTRACT FROM AUTHOR]

Interval-valued data have been widely used in both academic and industry research areas. With the development of computational statistics, the Bayesian methods are growing rapidly. Based on the existing theoretical researches, this paper tries to establish a Bayesian framework for interval-valued data, which contains the determination of Jeffreys' prior, posterior inferences, and posterior predictive checking methods. First, Jeffreys' prior for interval-valued data is proposed for the first time. Then, the posterior conditional distribution is derived, and posterior samples are obtained by using Gibbs sampling methods. Based on posterior samples, we obtain the Bayesian point and credible interval estimations for interval-valued data. Finally, we propose a posterior predictive checking method for interval-valued data based on novel test quantities. If the posterior predictive checking method shows that the data do not meet the distribution assumptions, we suggest the use of a data transformation. The effectiveness of our method is demonstrated by simulations and two cardiological data sets. [ABSTRACT FROM AUTHOR]