Methods of probability and imprecise probability for uncertainty quantification in applied problems

The quantification of uncertainty is a critical step in modeling systems and assessing their reliability. Traditionally, uncertainty in reliability has been represented by frequentist probability and more recently by Bayesian probability. Now in addition to these approaches, there are numerous imprecise methods available based on the generalization of probability. Each method, whether it be precise or imprecise, has different merits and deficits for uncertainty representation that can dramatically impact the analysis. Through this research, I endeavor to better understand the implications of the choice of uncertainty quantification method for applied problems. Towards this end, two informational aspects of uncertainty are investigated from the perspective of Bayesian probability and two generalized classes of probability. Specifically, how the amount of available information impacts the inference obtained from different models is studied; and can sets of probabilities enhance the ability to make simultaneous inferences with multilevel data on a multilevel system with different models.;These ideas are investigated in the context of two simplified applied problems. The first problem involves a biased coin flip experiment modeled by three precise Bayesian beta priors as well as two types of imprecise beta models (IBM). The cross entropy with the Kullback-Leibler divergence is used to compare the information contained in the precise and imprecise models. While the use of this measure is precedented with classical Bayesian probability, this is the first application of cross entropy for the comparison of sets of probabilities in the continuous domain.;The second problem expands on recent advances in inference for multilevel data in fault trees and Bayesian networks. Specifically, three a priori representations of uncertainty are considered for the parameters of a fault tree or Bayesian network: a multinomial-Dirichlet model with known hyperparameters, a multinomial-Dirichlet model with unknown hyperparameters, and an imprecise Dirichlet model (IDM). The a posteriori uncertainty after updating with data are calculated using Markov chain Monte Carlo and the results of the three approaches are discussed.